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ZHANG WENPENG
editor
SCIENTIA MAGNA
International Book Series
Vol. , No.
Editor
Dr. Zhang Wenpeng
Department of Mathematics
Northwest University
Xian, Shaanxi, P.R.China
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Contributing to Scientia Magna book series
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Associate Editors
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Technology, IIT Madras, Chennai , Tamil Nadu, India.
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P.R.China. Email xueyanrong.com
iii
Contents
N. Dung Notes Ponomarevsystems
Y. Cho, etc. QuasiPseudoMetrization in PqpMSpaces
S. Khairnar, etc. On Smarandache least common multiple ratio
R. Maragatham Some new relative character graphs
D. Senthilkumar and K. Thirugnanasambandam Class A weighted
composition operators
S. Yilmaz Position vectors of some special spacelike curves according to
Bishop frame
C. Chen and G. Wang Monotonicity and logarithmic convexity properties
for the gamma function
C. Prabpayak and U. Leerawat On ideas and congruences in KUalgebras
J. Tian and W. He Browders theorem and generalized Weyls theorem
M. Karacan and L. Kula On the instantaneous screw axes of two parameter
motions in Lorentzian space
Bin Zhao and Shunqin Wang Cyclic dualizing elements in Girard quantales
S. Balasubramanian, etc. Compact spaces
P. S. Sivagami and D. Sivaraj and sets of generalized Topologies
Y. Yayli and E. Tutuncu Generalized galilean transformations and dual
Quaternions
T. Veluchamy and P. Sivakkumar On fuzzy number valued Lebesgue
outer measure
D. Cui, etc. A note to Lagrange mean value theorem
L. Torkzadeh and A. Saeid Some properties of , fuzzy BGalgebras
Y. Yang and M. Fang On the Smarandache totient function and the Smarandache
power sequence
Y. Guo A new additive function and the F. Smarandache function
iv
Scientia Magna
Vol. , No. ,
Notes on Ponomarevsystems
Nguyen Van Dung
Mathematics Faculty, Dongthap University, Pham Huu Lau, Ward , Caolanh City,
Dongthap Province, Vietnam
Email nguyendungtcyahoo.com
Abstract Let f, M, X, P be a Ponomarevsystem. We prove that f is a sequencecovering
resp., pseudosequencecovering, subsequencecovering map i P is a strong csnetwork
resp., strong cs
network, pseudostrong cs
network for X. Moreover, subsequence
covering can be replaced by sequentiallyquotient, and pseudostrong cs
network can
be replaced by pseudostrong csnetwork. As applications of these results, we get many
wellknown results on images of metric spaces and more.
Keywords compact map, smap, csmap, strong smap, sequencecovering map, pseudo
sequencecovering map, sequentiallyquotient map, subsequencecovering map, Ponomarev
system
. Introduction
Spaces with pointcountable cfpnetworks resp., pointcountable csnetworks, pointcoun
table cs
networks can be characterized as simages of a metric space M under a coveringmap
f, and many results have been proved , , , . Recently, some authors have tried
to generalize these results , , . In , Y. Ge and J. Shen have obtained the following.
Theorem .. , Theorem . Let f, M, X, P be a Ponomarevsystem, then f is a
compactcovering map i P is a strong knetwork for X.
Take Theorem . into account, and note that compactcovering maps, sequencecovering
maps, pseudosequencecovering maps, subsequencecovering maps and sequentiallyquotient
maps have a closed relation , , , the following question naturally arises.
Question .. Let f, M, X, P be a Ponomarevsystem. What is the necessary and
sucient condition such that f is a sequencecovering pseudosequencecovering, subsequence
covering, sequentiallyquotient map
In this paper, we introduce denitions of strong csnetwork, strong cs
network, pseudo
strong csnetwork and pseudostrong cs
network as modications of strong knetwork in to
give necessary and sucient conditions such that f is sequencecovering resp.,
pseudosequence
covering, subsequencecovering, sequentiallyquotient where f, M, X, P is a Ponomarevsyste
m. As applications of these results, we get many wellknown results on images of metric
spaces
and more.
Throughout this paper, all spaces are assumed to be regular and T
, all maps are assumed
continuous and onto, N denotes the set of all natural numbers, denotes N and convergent
Nguyen Van Dung No.
sequence includes its limit point. Let f X Y be a map and P be a collection of subsets of
X, we denote
P
P P P and fP fP P P.
Denition .. Let A be a subset of a space X and P be a collection of subsets of X.
P is a network at x in X, if x P for every P P and whenever x U with U open
in X, then x P U for some P P.
P is a network for X, if whenever x U with U open in X, then x P U for some
P P.
P is a knetwork for X, if whenever K U with K compact and U open in X, then
K
F U for some nite F P.
P is a cfpcover for A in X, if P is a cover for A in X such that it can be precisely
rened by some nite cover consisting of closed subsets of A.
P is a cfpnetwork for A in X, if whenever K is a compact subset of A and K U
with U open in X, there exists a subfamily F of P such that F is a cfpcover for K in A and
F U. A such P in is called to have property cc for A.
P is a strong knetwork for X, if whenever K is a compact subset of X, there exists a
countable subfamily P
K
of P such that P
K
is a cfpnetwork for K in X. Note that there is a
dierent denition of strong knetwork in .
P is a csnetwork for A in X, if whenever S is a convergent sequence in A converging
to x A U with U open in X, then S is eventually in P U for some P P.
P is a cs
network for A in X, if whenever S is a convergent sequence converging to
x A U with U open in X, then S is frequently in P U for some P P.
P is compactcountable, if P P K P is countable for each compact subset
K of X.
P is pointnite, if P P x P is nite for each point x X.
P is locally nite, if for each point x X, there is a neighborhood U of X such that
U meets only nitely many members of P.
P is locally nite, if P
P
n
n N where each P
n
is a locally nite collection.
P is pointcountable, if P P x P is countable for each point x X. Note
that every pointnite locally countable, compactcountable collection is a pointcountable
collection.
If A X, then a csnetwork resp., cs
network, cfpcover, cfpnetwork for A in X
is called a csnetwork resp., cs
network, cfpcover, cfpnetwork for X see , .
Denition .. Let f X Y be a map.
f is a compactcovering map, if each compact subset of Y is the image of some compact
subset of X.
f is an smap, if whenever y Y , then f
y is a separable subset of X.
f is a strong smap, if for each y Y , there exists a neighborhood V of y in Y such
that f
V is a separable subset of X.
f is a csmap, if whenever K is a compact subset of Y , then f
K is a separable
subset of X.
f is a compact map, if whenever y Y , then f
y is a compact subset of X.
Vol. Notes Ponomarevsystems
f is a map, if there is a base B for X such that fB is a locally nite collection of
subsets of Y .
f is a sequencecovering map, if each convergent sequence in Y is the image of some
convergent sequence in X.
f is a pseudosequencecovering map, if each convergent sequence in Y is the image of
some compact subset of X.
f is a sequentiallyquotient map, if for each convergent sequence S in Y , there is a
convergent sequence L in X such that fL is a subsequence of S.
f is a subsequencecovering map, if for each convergent sequence S in Y , there is a
compact subset K of X such that fK is a subsequence of S.
Denition .. Let X be a space.
X is an
space, if X has a countable csnetwork. Note that cs can be replaced by
k or cs
.
X is an space, if X has a locally nite csnetwork. Note that cs can be replaced
by k or cs
.
X is a Frechet space, if whenever x A with A X, then there is a sequence in A
converging to x.
X is a sequential space, if whenever A is a non closed subset of X, then there is a
sequence in A converging to a point not in A.
Denition .. Let P be a network for a space X. Assume that P is closed under
nite intersections, and there exists a countable subfamily P
x
of P such that P
x
is a network
at x in X for every x X. Put P P
. For every n N, put
n
and endowed
n
with the discrete topology. Put
M
b
n
nN
n
P
n
n N forms a network at some point x
b
X
.
Then M, which is a subspace of the product space
nN
n
, is a metric space and x
b
is
unique for every b M. Moreover x
b
nN
P
n
. Dene f M X by fb x
b
, then f is a
map and f, M, X, P is called a Ponomarevsystem.
Remark .. If P is a pointcountable network for X, then P
x
PPxPP
is a countable network at x in X for every x X. It implies that the Ponomarevsystem
f, M, X, P exists.
For terms which are not dened here, please refer to and
. Results
In the following, f, M, X, P denotes a Ponomarevsystem where f and M are dened in
Denition ..
Firstly we introduce some denitions.
Denition .. Let P be a csnetwork for a convergent sequence S in X where S U
with U open in X, then a subfamily F of P has property csS, U if it satises the following.
Nguyen Van Dung No.
F is nite,
F S F U for every F F,
If x S, then there is a unique F F such that x F,
If F F contains the limit point of S, then S F is nite.
The following lemma proves that a family having property csS, U exists.
Lemma .. If P is a csnetwork for a convergent sequence S in X where S U with U
open in X, then there is a subfamily F of P such that F has property csS, U.
Proof. Let S x
m
m converging to x
X. Since P is a csnetwork for S
in X, there is P
P such that S is eventually in P
and P
U. Because S U is nite,
SU x
m
i
i , , k for some k N. For every i , , k, note that U
Sx
m
i
is an open neighborhood of x
m
i
in X, so there is P
i
P such that x
m
i
P
i
U
Sx
m
i
.
Put F P
i
i , , k, then F satises required conditions.
Denition .. Let P be a cs
network for a convergent sequence S in X where S U
with U open in X, then a subfamily F of P has property cs
S, U if it satises the following.
F is nite,
F S F U for every F F,
If x S, then there is some F F such that x F,
F S is closed for every F F.
The following lemma proves that a family having property cs
S, U, if P is pointcountable.
Lemma .. If P is a pointcountable cs
network for a convergent sequence S in X
where S U with U open in X, then there is a subfamily F of P such that F has property
cs
S, U.
Proof. Let S x
n
n converging to x
X. Since P is pointcountable cs
network for S in X, P P x
P U is nonempty and countable. Put P P x
PUP
i
i N. We shall show that there exists an n
N such that x
n
n
i
P
i
for all
but nitely many n N. If not, we can choose a subsequence x
n
k
k N of x
n
n N as
follows.
x
n
SP
P
n
for some P
n
P,
x
n
k
S
n
k
i
P
i
P
n
k
for some P
n
k
P, for all k gt .
So each P
i
only includes nitely many elements of x
n
k
k N. But P is a cs
network for S in
X, then there is P P such that x
x
n
k
k N is frequently in P. Thus P P
m
for some
m N. Hence P
m
includes innitely many elements of x
n
k
k N, a contradiction. Then we
can put S
n
i
P
i
x
n
i
i , , k for some k N. For each i , , k there is F
i
P
such that x
n
i
F
i
USx
n
i
. Put F F
i
i,,kP
i
i,,n
,
then F satises required conditions.
The following notions are modications of the strong knetwork in .
Denition .. Let P be a collection of subsets of a space X.
P is a strong csnetwork for X, if whenever S is a convergent sequence in X, then there
exists a countable subfamily P
S
of P such that P
S
is a csnetwork for S in X.
Vol. Notes Ponomarevsystems
P is a strong cs
network for X, if whenever S is a convergent sequence in X, then
there exists a countable subfamily P
S
of P such that P
S
is a cs
network for S in X.
P is a pseudostrong csnetwork for X, if whenever S is a convergent sequence in X,
then there exists a countable subfamily P
S
of P such that P
S
is a csnetwork in X for some
convergent subsequence T of S.
P is a pseudostrong cs
network for X, if whenever S is a convergent sequence in X,
then there exists a countable subfamily P
S
of P such that P
S
is a cs
network in X for some
convergent subsequence T of S.
Remark .. We have following implications from the above denitions.
Strong csnetwork csnetwork.
Strong csnetwork strong cs
network pseudostrong csnetwork pseudostrong
cs
network cs
network.
The following example proves that some inverse implications in Remark . do not hold.
Example .. There exists a Ponomarevsystem f, M, X, P such that P is a csnetwork
for X, but P is not a strong csnetwork for X.
Proof. Let X be the sequential fan space S
, then X has not any countable base at
x
, where x
is the nonisolated point in X. Put P U X U is open in X
x
,
then P is a network for X, and there exists a countable subfamily P
x
of P such that P
x
is a
network at x in X for every x X. Therefore the Ponomarevsystem f, M, X, P exists.
Since P contains a base of X, P is a csnetwork for X. We shall prove that P is not a
strong csnetwork for X. Let S be a nontrivial convergent sequence converging to x
. If P
is a strong csnetwork for X, there exists a countable subfamily P
S
of P such that P
S
is a
csnetwork for S in X. Note that every element of P
S
x
is open in X and P
S
x
is a countable network at x
in X. Therefore P
S
x
is a countable neighborhood base at
x
in X. This contradicts that X has not any countable neighborhood base at x
. It implies
that P is not a strong csnetwork for X.
Remark .. Similarly, we get that there exists a Ponomarevsystem f, M, X, P such
that P is a cs
network for X, but P is not a pseudostrong cs
network for X where X and
P are the same in example .. Moreover P is not neither a pseudostrong csnetwork for X
nor a strong cs
network for X by remark .. But we dont know whether remaining inverse
implications in remark . do not hold.
The following lemma establishes a equivalent condition between a strong csnetwork and
a csnetwork.
Lemma .. If P is a pointcountable family consisting of subsets of a space X, then the
following are equivalent.
P is a strong csnetwork for X,
P is a csnetwork for X.
Proof. . By Remark ..
. Let S be a convergent sequence in X. Since P is pointcountable, then P
S
P P P S is countable. Obviously P
S
is a csnetwork for S in X. It implies that P
is a strong csnetwork for X.
Moreover we get that.
Nguyen Van Dung No.
Lemma .. Let P be a pointcountable csnetwork for X. If each compact subset of X
is rstcountable, then P is a cfpnetwork for X.
Proof. Let K be a compact subset of X and K U with U open in X. It follows from
Lemma . in that there are P
x
,Q
x
P such that x int
K
Q
x
KG
x
cl
K
G
x
int
K
P
x
KP
x
U with some G
x
open in K for every x K. Since K is compact, there is
a nite subset F of K such that G
x
x F covers K. Then F P
x
x F is a cfpcover
for K in X with
F U. It implies that P is a cfpnetwork for X.
We also have a equivalent condition for a strong cs
network, pseudostrong cs
network
and a cs
network as follows.
Lemma .. If a family P is a pointcountable family consisting of subsets of a space
X, then the following are equivalent.
P is a strong cs
network for X ,
P is a pseudostrong cs
network for X,
P is a cs
network for X.
Proof. . By Remark ..
. Let S be a convergent sequence in X. Since P is pointcountable, then P
S
P P P S is countable. Obviously P
S
is a cs
network for S in X. It implies that
P is a strong cs
network for X.
Regarding the relations between coveringmaps we have the following.
Lemma .. , Remark . Let f X Y be a map.
If f is quotient and X is sequential, then f is sequentiallyquotient.
If Y is sequential and f is sequentiallyquotient, then f is quotient.
Lemma .. , Remark Let f X Y be a map.
If f is compactcovering or sequencecovering, then f is pseudosequencecovering.
If f is pseudosequencecovering or sequentiallyquotient, then f is subsequencecovering.
Moreover we get that.
Lemma .. Let f X Y be a map. If X is sequential and f is subsequencecovering,
then f is sequentiallyquotient.
Proof. Let S be a convergent sequence converging to a point y Y . Since f is
subsequencecovering, there is a compact subset K in X such that fK is a convergent subse
quence of S. Put fK yy
n
n N where y
n
n N converges to y. For each n N
pick x
n
f
y
n
K, then x
n
n N K. Note that K is a compact subset in a sequential
space, K is sequentially compact. So there is a convergent subsequence x x
n
k
k N of
xx
n
n N that converges to x f
y. Then y fx
n
k
k N is a convergent
subsequence of yy
n
n N. Therefore yfx
n
k
k N is a convergent subsequence
of S. This proves that f is sequentiallyquotient.
Now we establish a relation between covers and map in a ponomarevsystem.
Lemma .. Let f, M, X, P be a Ponomarevsystem, then the following hold.
f is a compact map i P is pointnite.
f is an smap i P is pointcountable.
f is a csmap i P is compactcountable.
f is a strong smap i P is locally countable.
Vol. Notes Ponomarevsystems
Proof. and . By Proposition . in .
Necessary. Conversely, if P is not compactcountable, then there exists some compact
subset K of X such that P
K is uncountable. Let x K and P
n
nN
is a network at x in X. For every put c
n
where
and
n
n
for n gt .
Then P
n
n N is a network at x, so c
f
K. Put U
c
n
M
for every . We shall prove that U
is an open cover for f
K in M. Note
that every U
is open and nonempty, and if c
n
f
K, then P
n
n N P is
a network at fc K. It implies that P
K . Then
for some . Hence
U
is an open cover for f
K in M. Since f is a csmap, f
K is separable in
M. Then U
has a countable subcover. It is a contradiction because U
U
whenever .
Suciency. Let K be a compact subset of X. Since P is compactcountable,
n
n
P
K is countable for every n N. Then f
K
nN
n
. Since
nN
n
is a
hereditarily separable space, it implies that f
K is separable in M, i.e, f is a csmap.
It is similar to the proof of .
From Theorem . in see Theorem . the authors have obtained the following.
Corollary .. , Proposition . Let f, M, X, P be a Ponomarevsystem, then the
following are equivalent.
f is a compactcovering smap,
P is a pointcountable strong knetwork,
P is a pointcountable cfpnetwork.
Note that s and pointcountable in Corollary . can be replaced by compact and
pointnite respectively , Remark .. Using Lemma . we get that s and point
countable in Corollary . can be also replaced by cs and compactcountable, or strong
s and locally countable respectively. Moreover, the following holds.
Corollary .. , Corollary The following are equivalent for a space X.
X is a compactcovering simage of a metric space,
X has a pointcountable cfpnetwork.
Proof. . Let f M X be a compactcovering smap from a metric space M
onto X. Since M is metric, M has a locally nite base B. Then fB is a pointcountable
cfpnetwork for X.
. Let P be a pointcountable cfpnetwork for X. It follows from Remark . that
the Ponomarevsystem f, M, X, P exists. Then f is a compactcovering smap by Corollary
.. It implies that X is a compactcovering simage of a metric space.
Next we give a technical lemma which plays an important role in the following parts.
Lemma .. Let f, M, X, P be a Ponomarevsystem, b
n
M where P
n
n
N is a network at some x
b
X and
U
n
c
i
M
i
i
for all i n,
for every n N. Then we have.
U
n
n N is a base at b in M.
Nguyen Van Dung No.
fU
n
n
i
P
i
for every n N.
Proof. . By denition of the product topology of a countable family consisting of
discrete spaces.
. For each n N, let x fU
n
. Then x fc for some c
i
U
n
. It implies that
x
iN
P
i
n
i
P
i
n
i
P
i
, i.e, fU
n
n
i
P
i
.
Conversely, let x
n
i
P
i
. Then x fc with c
i
M. Note that for each i N
there exists some
ni
ni
such that
ni
i
. Put d
i
with
i
i
for all i n.
Then we get d U
n
and fd x. It implies that
n
i
P
i
fU
n
.
By the above we get fU
n
n
i
P
i
.
Now we give a necessary and sucient condition such that f is a sequencecovering map.
Theorem .. Let f, M, X, P be a Ponomarevsystem, then the following are equiva
lent.
f is a sequencecovering map,
P is a strong csnetwork for X.
Proof. . Let S be a convergent sequence in X, we shall prove that there is
a countable subfamily P
S
of P such that P
S
is a csnetwork for S in X. Indeed, since f
is sequencecovering, there exists a convergent sequence C in M such that S fC. Put
p
n
C n N, where every p
n
iN
i
n
is a projection, then is countable. Let
P
S
P
, is nite
.
Since P is closed under nite intersections, P
S
is a countable subfamily of P. It suces to
show that P
S
is a csnetwork for S in X. Let L be a convergent sequence in S converging to
x U with U open in X. Since f is sequencecovering, there exists a convergent subsequence
T of C in M such that fT L. We get that T converges to some b f
xf
U. Let
b
n
. For each n N put
U
n
c
i
M
i
i
for all i n.
It follows from Lemma . that U
n
n N is a base at b in M. Since T converges to
bf
U which is open in M, T is eventually in some U
n
with U
n
f
U. Therefore L is
eventually in fU
n
U. Since fU
n
n
i
P
i
by Lemma . and
n
i
P
i
is an element of
P
S
, we get that P
S
is a csnetwork for S in X.
. Let S x
m
m be a convergent sequence in X converging to x
. We
shall prove that S fC for some convergent sequence C in M. Assume that all x
m
s are
distinct. Since P is a strong csnetwork for X, there exists a countable subfamily P
S
of P such
that P
S
is a csnetwork for S. It follows from Lemma . that there exists a subfamily F of P
S
Vol. Notes Ponomarevsystems
such that F has property csS, X. Since P
S
is countable, F P
S
F has property csS, X
is countable by niteness of F. Put
F P F has property csS, X F
n
n N,
and put F
n
P
n
for every n N where
n
is a nite subset of
n
. For every
m and n N, since F
n
has property csS, X, there is a unique
n,m
n
such that
x
m
P
n,m
F
n
. Put b
m
n,m
nN
n
and C b
m
m , we shall prove that C is
a convergent sequence in M and fC S.
To show C M and fC S it suces to prove that P
n,m
n N is a network in X
at x
m
for every m . Indeed, let U be an open neighborhood of x
m
in X. We consider two
following cases.
ax
m
x
.
We get that U S is a convergent sequence in X and S U U. It follows from
Lemma . that there is a subfamily F of P
S
such that F to have property csS U, U. Since
S S U is nite, put S S U x
m
i
i , , l. For every i , . . . , l, note
that X S x
m
i
is an open neighborhood of x
m
i
in X, so there is P
i
P
S
such that
x
m
i
P
i
XSx
m
i
. It is easy to see that FP
i
i , , l has property csS, X.
So there is k N such that F P
i
i,,lF
k
. Thus x
P
k,
F
k
. Note that
P
k,
must be an element of F which has property csS U, U. It implies that x
P
k,
U.
bx
m
x
.
We get that S x
m
is a convergent sequence in X and S x
m
Xx
m
with
Xx
m
open. It follows from Lemma . that there exists a subfamily F of P
S
such that
F has property csS x
m
,Xx
m
. Note that U S x
m
is an open neighborhood
of x
m
, so there exists P
m
P
S
such that x
m
P
m
USx
m
. It is easy to see
that F P
m
has property csS, X. Hence there is k N such that F P
m
F
k
, then
x
m
P
k,m
P
m
U.
By the above we get that x
m
P
k,m
U for every m . Then P
n,m
n N is a
network in X at x
m
for every m . It implies that C M and fC S. To complete
the proof we shall prove that C b
m
m converging to b
. For every k N there is
a unique
k,
k
such that x
P
k,
F
k
. Since F
k
has property csS, X, S P
k,
is
nite. So there is m
k
N such that x
m
P
k,
for every m gt m
k
. Note that x
m
P
k,m
F
k
.
Thus
k,m
k,
for every m gt m
k
. So
k,m
m converging to
k,
as m . Hence
Cb
m
m is a convergent sequence in M converging to b
. It implies that S fC
with C being a convergent sequence in M, i.e, f is a sequencecovering map.
By Theorem . we get nice characterizations of simages of metric spaces which were
obtained in as follows.
Corollary .. Theorem . The following are equivalent for a space X.
X is a sequencecovering simage of a metric space,
X has a pointcountable csnetwork.
Proof. . Let f M X be a sequencecovering smap from a metric space M
onto X. Since M is metric, M has a locally nite base B. Then fB is a pointcountable
csnetwork for X.
Nguyen Van Dung No.
. Let P be a pointcountable csnetwork for X. It follows from Remark . that
the Ponomarevsystem f, M, X, P exists.
By Lemma ., P is a strong csnetwork for X. From Theorem . and Lemma . we
get that f is a sequencecovering smap. It implies that X is a sequencecovering simage of a
metric space.
Corollary .. , Theorem . The following are equivalent for a space X.
X is a sequencecovering, compactcovering quotient simage of a metric space,
X is a sequential space having a pointcountable csnetwork.
Proof. . From Corollary ., we only need to prove that X is sequential. It is
clear because f is a quotient map from a metric space onto X.
. Let P be a pointcountable csnetwork for X. As in the proof of Corollary .,
then X is an image of a metric space M under a sequencecovering smap f where f, M, X, P
is a Ponomarevsystem. It follows from Lemma ., Lemma . and Lemma . that f is
quotient. We shall prove that f is compactcovering by showing that P is a pointcountable
cfpnetwork for X. Let K be a compact subset of X. Since X is sequential, K is sequential
compact. From Proposition . in , P
K
P K P P is a pointcountable knetwork
for K. It follows from Theorem . in that K is metrizable. Then P is a pointcountable
cfpnetwork for X by Lemma .. From Corollary ., f is compactcovering.
The following is routine.
Corollary .. The following are equivalent for a space X.
X is a sequencecovering, compactcovering pseudoopen simage of a metric space,
X is a Frechet space having a pointcountable csnetwork.
Moreover we get a mapping theorem on spaces which belongs to .
Corollary .. , Theorem The following are equivalent for a space X.
X is an space,
X is a sequencecovering, compactcovering image of a metric space,
X is a compactcovering image of a metric space,
X is a sequencecovering image of a metric space,
X is a pseudosequencecovering image of a metric space,
X is a subsequencecovering image of a metric space,
X is a sequentiallyquotient image of a metric space.
Proof. . Since X is an space, X has a locally nite csnetwork Q
Q
n
n N and a locally nite knetwork R
R
n
n N where every Q
n
and R
n
is locally
nite and every elements of Q and R are closed. For each n N put P
n
Q
n
R
n
, then
P
n
is locally nite. Therefore X has a locally nite cs and knetwork P
P
n
n N.
It follows from Remark . that the Ponomarevsystem f, M, X, P exists. We shall prove
that f is a sequencecovering and compactcovering map. From Lemma ., P is a strong
csnetwork for X. Hence f is a sequencecovering map by Theorem .. Since P is a locally
nite closed knetwork, P is a cfpnetwork. Hence f is a compactcovering map by Corollary
.. To complete the proof it suces to show that f is a map. For each b
b
n
M where
P
b
n
n N is a network at some x
b
in X, put U
b
n
c
i
M
i
b
i
for all i n,
for each n N and B
b
U
b
n
n N. It follows from Lemma . that B
B
b
bM
Vol. Notes Ponomarevsystems
is a base for M and fU
b
n
n
i
P
b
i
. Since P is locally nite, fB is a locally nite. It
implies that f is a map.
, . Obviously.
, , . From Lemma ..
. From Lemma ..
. Let f M X be a sequentiallyquotient map from a metric space M onto
X. Since f is a map, M has a base B such that fB is a locally nite collection of subsets
of X. On the other hand f is sequentiallyquotient, fB is a locally nite cs
network for
X. It implies that X is an space.
Next we give a necessary and sucient condition such that f is a pseudosequencecovering
map.
Theorem .. Let f, M, X, P be a Ponomarevsystem, then the following are equiva
lent.
f is a pseudosequencecovering map,
P is a strong cs
network for X.
Proof. . Let S be a convergent sequence in X, we shall prove that there is
a countable subfamily P
S
of P such that P
S
is a cs
network for S in X. Indeed, since f is
pseudosequencecovering, there exists a compact subset K in M such that S fK. Put
p
n
K n N, where every p
n
nN
n
n
is a projection, then is countable. Let
P
S
P
, is nite
.
Since P is closed under nite intersections, P
S
is a countable subfamily of P. We shall prove
that P
S
is a cs
network for S in X. Let L be a convergent sequence in S converging to
x S U with U open in X. As in the proof of Lemma ., there exists a convergent
subsequence T K in M such that fT is a convergent subsequence of L. We get that T
converges to b f
xf
U. Let b
n
. For each n N put
U
n
c
i
M
i
i
for all i n.
It follows from Lemma . that U
n
n N is a base at b in M. Since T converges to
bf
U which is open in M, T is eventually in some U
n
with U
n
f
U. Therefore L
is eventually in fU
n
U. From Lemma . fU
n
n
i
P
i
, and moreover
n
i
P
i
is an
element of P
S
, we get that P
S
is a cs
network for S in X.
. Let S x
m
m be a convergent sequence in X converging to x
. We
shall prove that S fK for some compact subset K in M. Assume that all x
m
s are distinct.
Since P is a strong cs
network for X, there exists a countable subfamily P
S
of P such that P
S
is a cs
network for S. It follows from Lemma . that there exists a subfamily F of P
S
such
that F has property cs
S, X. Since P
S
is countable, F P
S
F has property cs
S, X is
countable by niteness of F. Put
F P F has property cs
S, X F
n
n N,
Nguyen Van Dung No.
and put F
n
P
n
for every n N where
n
is a nite subset of
n
.
Put
K
b
n
nN
n
nN
P
n
S
.
Then we have that.
a K M and fK S.
Let b
n
K, then
nN
P
n
S . Thus there is x
m
S such that x
m
nN
P
n
.
We shall prove that P
n
n N is a network at x
m
in X. Indeed, let U be an any open
neighborhood of x
m
in X. We consider two following cases.
ix
m
x
.
We get that U S is a convergent sequence in X and U S U. It follows from
Lemma . that there exists a subfamily F of P
S
such that F has property cs
U S, U. Since
S U S is nite, S U S x
n
i
i , , l for some l N. For every i , , l,
note that X
Sx
n
i
is an open neighborhood for x
n
i
in X, there exists P
i
P
S
such
that x
n
i
P
i
XSx
n
i
. It is easy to see that F P
i
i , . . . , l has property
cs
S, X. So there exists k N such that F P
i
i,...,lF
k
. Since x
P
i
for
all i , . . . , l and x
P
k
F
k
, then P
k
F. Note that F has property cs
S U, U,
P
k
U. It implies that x
m
P
k
U.
ii x
m
x
.
We get that S x
m
is a convergent sequence in X and S x
m
Xx
m
with
Xx
m
open. It follows from Lemma . that there exists a subfamily F of P
S
such that F
has property cs
Sx
m
,Xx
m
. Note that
USx
m
is an open neighborhood
of x
m
, so there exists P
m
P
S
such that x
m
P
m
USx
m
U. It is easy to see
that F P
m
has property cs
S, X. So there exists k N such that F P
m
F
k
. Since
x
m
F for every F F and x P
k
F
k
, then x
m
P
k
P
m
U.
From the above arguments i and ii we get that P
n
n N is a net work at x
m
for
every m . It implies that b M and fb x
m
S, i.e, K M and fK S.
b S fK.
Let x S. For every n N, pick
n
n
such that x P
n
, then x
nN
P
n
S.
Put b
n
, then b K. Using the argument in a we get x fb. It implies that S fK.
c K is a compact subset of M.
Since K is a subset of
nN
n
and
nN
n
is a compact subset by niteness of each
n
, it
suces to prove that K is closed in
nN
n
. Let b
n
nN
n
K, then
nN
P
n
S.
Note that P
n
S is closed by property cs
S, X of F
n
for every n N . Since S is compact
in X and
nN
P
n
S,
n
n
P
n
S for some n
N. Put
W
c
n
nN
n
n
n
if n n
,
then W is an open neighborhood of b in
nN
n
with W K . So K is a closed subset of
Vol. Notes Ponomarevsystems
nN
n
. It implies that K is compact.
From a, b and c we get that S fK with K compact in M. It implies that f is a
pseudosequencecovering map.
In the following part, we give a necessary and sucient condition such that f is a subsequence
covering map or a sequentiallyquotient map.
Theorem .. Let f, M, X, P be a Ponomarevsystem, then the following are equiva
lent.
f is a subsequencecovering map,
f is a sequentiallyquotient map,
P is a pseudostrong csnetwork for X,
P is a pseudostrong cs
network for X.
Proof. . From Lemma ..
. Let S be a convergent sequence in X, we shall prove that there is a countable
subfamily P
S
of P such that P
S
is a csnetwork in X for some subsequence T of S. Indeed,
since f is sequentiallyquotient, there exists a convergent sequence C in M such that T fC
is a convergent subsequence of S. Put
p
n
C n N, where every p
n
nN
n
n
is a projection, then is countable. Let
P
S
P
, is nite
.
Since P is closed under nite intersections, P
S
is a countable subfamily of P. As in the proof
of Theorem . we get that P
S
is a csnetwork for T in X. It implies that P is a
pseudostrong csnetwork for X.
. From Remark ..
. Let S be a convergent sequence in X, we shall prove that there is a compact
subset K in M such that fK is a convergent subsequence of S. Indeed, since P is a pseudo
strong cs
network for X, there exists a countable subfamily P
S
of P such that P
S
is a cs
network in X for some convergent subsequence T of S. As in the proof of Theorem
. we get that there is a compact subset K in M such that fK T. It implies that f is a
subsequencecovering map.
By Theorem . and Theorem . we get a nice characterization of pseudosequence
covering subsequencecovering, sequentiallyquotient simages of metric spaces as follows.
Corollary .. The following are equivalent for a space X.
X is a pseudosequencecovering simage of a metric space,
X is a subsequencecovering simage of a metric space,
X is a sequentiallyquotient simage of a metric space,
X has a pointcountable cs
network.
Proof. . From Lemma ..
. From Lemma ..
. Let f M X be a sequentiallyquotient smap from a metric space M
onto X. Since M is metric, M has a locally nite base B. Then fB is a pointcountable
cs
network for X.
Nguyen Van Dung No.
. Let P be a pointcountable cs
network for X. It follows from Remark . that
the Ponomarevsystem f, M, X, P exists. By Lemma ., P is a strong cs
network for X.
From Theorem . and Lemma . we get that f is a pseudosequencecovering smap. It
implies that X is a pseudosequencecovering simage of a metric space.
From Corollary . we get a nice characterization of pseudosequencecovering quotient
simages quotient simages of metric spaces due to and as follows.
Corollary .. , Theorem . and , Theorem . The following are equivalent
for a space X.
X is a pseudosequencecovering quotient resp., pseudoopen simage of a metric
space,
X is a quotient resp., pseudoopen simage of a metric space,
X is a sequential resp., Frechet space having a pointcountable cs
network.
Finally we consider a particular case when M is separable.
Lemma .. Let f, M, X, P be a Ponomarevsystem, then the following are equivalent.
M is separable,
P is countable.
Proof. . If P is not countable, then is uncountable. For each put
U
c
n
M
. Then each U
is a nonempty open subset of M. We shall
prove that U
covers M. Indeed, if c
n
M, then P
n
n N P is a
network at fc. Pick
then c U
. It implies that U
is an open
cover for M. Since M is separable, U
has a countable subcover. It is a contradiction
because U
U
whenever .
. Obviously.
From the above we get a mapping theorem on
spaces which belongs to .
Corollary .. , Theorem The following are equivalent for a space X.
X is an
space,
X is a sequencecovering, compactcovering image of a separable metric space,
X is a compactcovering image of a separable metric space,
X is a sequencecovering image of a separable metric space,
X is a pseudosequencecovering image of a separable metric space,
X is a subsequencecovering image of a separable metric space,
X is a sequentiallyquotient image of a separable metric space.
Proof. . Since X is an
space, X has a countable csnetwork Qand a countable
knetwork R. Note that all elements of Q and R can be chosen closed. Put P QR, then P
is a countable cs and knetwork for X. It follows from Remark . that the Ponomarevsystem
f, M, X, P exists. Since P is countable, M is separable by Lemma .. From Lemma ., P
is a strong csnetwork for X. Hence f is a sequencecovering map by Theorem .. On the
other hand, P is a countable closed knetwork, P is a cfpnetwork for X. Hence f is a compact
covering map by Corollary .. It implies that f is a compactcovering and sequencecovering
map from a separable metric space M onto X.
, . Obviously.
, , . From Lemma ..
Vol. Notes Ponomarevsystems
. From Lemma ..
. Let f M X be a sequentiallyquotient map from a separable metric space
M onto X. Since M is separable metric, M has a countable base B. Then fB is a countable
cs
network for X. It implies that X is an
space.
Remark .. If one of the above results contains pointcountable and simage, then
it is possible to replace them by pointnite and compact image, compactcountable
and csimage, or locally countable and strong simage respectively. Therefore we get
characterizations of compact images of metric spaces, characterizations of csimages of
metric
spaces, and characterizations of strong simages of metric spaces.
References
H. Chen, Compactcovering maps and knetworks, Proc. Amer. Math. Soc., ,
.
R. Engelking, General topology, Sigma series in pure mathematics, , Heldermann
Verlag, Berlin.
Y. Ge, Characterizations of snmetrizable spaces, Publ. Inst. Math. BeogradN.S.,
, No., .
Y. Ge,
spaces and images of separable metric sapces, Siberian Elec. Math. Rep.,
,.
Y. Ge and J. Shen, Networks in Ponomarevsystems, Sci. Ser. A, Math. Sci., ,
.
G. Gruenhage, E. Michael, and Y. Tanaka, Spaces determined by pointcountable covers,
Pacic J. Math., , No., .
Z. Li, A mapping theorem on spaces, Mat. Vesnik., , .
S. Lin, A note on the Arens space and sequential fan, Topology Appl., ,
.
S. Lin and C. Liu, On spaces with pointcountable csnetworks, Topology Appl., ,
.
S. Lin, C. Liu, and M. Dai, Images on locally separable metric spaces, Acta Math.
Sinica N.S., , No., .
S. Lin and P. Yan, Sequencecovering maps of metric spaces, Topology Appl., ,
.
S. Lin and P. Yan, Notes on cfpcovers, Comment. Math. Univ. Carolin, ,
No., .
Y. Tanaka, Pointcountable covers and knetworks, Topology Proc., , .
Y. Tanaka, Theory of knetworks II, Questions Answers in Gen. Topology, ,
.
Y. Tanaka and Z. Li, Certain coveringmaps and knetworks, and related matters,
Topology Proc., , No., .
Scientia Magna
Vol. , No. ,
QuasiPseudoMetrization in PqpMSpaces
Yeol Je Cho
, Mariusz T. Grabiec
and Reza Saadati
Department of Mathematics Education and the RINS, College of Education
Gyeongsang National University, Chinju , Korea
Department of Operation Research, Academy of Economics al. Niepodleglosci ,
Pozna n, Poland
Faculty of Science, University of Shomal, Amol, P.O. Box , Iran
Email yjchognu.ac.kr, m.grabiecae.poznan.pl, rsaadatieml.cc
Abstract. A bitopologiacl space X, T
P
,T
Q
generated by probabilisticquasipseudometric
P is quasipseudometrizable if and only if there exists a quasipseudometric which induces
those two topologies. In this paper, we consider a problem of quasipseudometrization of
PqpMspaces.
Keywords Probabilisticquasipseudometricspace PqpMspace, quasipseudometrization,
bitopological space X, T
P
,T
Q
, quasiuniform structure.
. Introduction
A topological space X, is said to be metrizable if there exists a metric d X
R
which generates a topology T
d
equivalent to .
For PqpMspaces X, P, , the problem leads to nding a quasipseudometric p X
R
such that it induces a topology T
p
equivalent to T
P
and its conjugate quasipseudometric q
generates a topology T
q
equivalent to T
Q
. In other words, a bitopological space X, T
P
,T
Q
is
quasipseudometrizable if and only if there exists a quasipseudometric p which induces those
two topologies.
In section , we dene the socalled PEspaces and show their properties. Also, we show
that a PEspace generates a quasiuniform structure on the underlying set cf., Fletcher and
Lindgren , which is quasipseudometrizable.
Next, we establish a number of relationships between PE and PqpMspaces.
In section , some conditions for the quasipseudometrizability of a PqpMspace are given.
Finally, in section , we consider the problem of pseudometrization in PqpMspaces.
. Preliminaries
We shall now consider functions dened on the extended real line R with values in the
complete lattice I, . Note that the family I
R
, is also a complete lattice. The following
Vol. QuasiPseudoMetrization in PqpMSpaces
subsets of I
R
will be used in the sequel
RFI
R
F is nondecreasing, leftcontinuous on R,
F and F ,
UR u
a
Ru
a
a,
for any a R.
Note that u
is the greatest element in R and u
is the smallest in it. Furthermore,
we dene
RFRF,
RF
R lim
t
Ft .
The family
R is a sublatices at R with the bounds u
and u
.
Denition .. Let S, , , be a set partially ordered with bounds and . A function
S
S is called a t
S
norm it the following condition holds for all a, b, c, d S,
S a b b a,
S a a,
S a b c a b c,
S a b c d whenever a c and b d.
Let TS, denote the family of all t
S
norm on the set S. Then the relation dened
by the formula
a
ba
b, a, b S, .
is a partially order in the family TS, .
Second relation in the family TS, is dened as follows
a
c
b
da
b
c
d, a, b, c, d S.
By putting b c in ., we obtain a
da
d for a, b S and hence
follows. Thus we know that
x
implies
.
According to the Denition ., a t
I
norm see and T I
I is in interval
I , an abelian semigroup with unit, and the t
I
norm T is nondecreasing with respect to
each variable.
Denition .. Let T be a t
I
norm.
T is called a continuous t
I
norm if the function T is continuous with respect to the
product topology on the set I I.
T is said to be leftcontinuous if, for every x, y , , the following holds
Tx, y supTu, v lt u lt x, lt v lt y.
T is said to berightcontinuous if, for every x, y , , the following conditions holds
Tx, y supTu, v x lt u lt , y, v lt .
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
We shall now establish the notation related to a few most important t
I
norm dened in
Denition .
Mx, y minx, y, for all x, y I. M is continuous and positive.
x, y x y, x, y I. is strictly increasing and continuous.
Wx, y maxx y , for all x, y I. W is continuous t
I
norm.
Zx, y
x, if x I and y I,
y, if x and y I,
, if x, y , .
The function Z is rightcontinuous, but it fails to be leftcondition. We give the following
relations among t
I
norms dened above
M W Z, .
MWZ.
We shall now present some properties of the t
S
norm dened on
R. According to
Denition .. the ordered pair
, is an abelian semigroup with the unit u
and the
operation
is a nondecreasing function. We notes that u
is a zero
. Indeed, by S and S, we obtain
u
u
Fu
u
u
,F
.
Let T be a leftcontinuous t
I
norm. Then the functiona
dened by
TF, Gt TFt, Gt, t , .
is a t
norm on the set
.
For every t
norm , the following inequality holds M, where M is the t
I
norm of
.
If T is leftcontinuous t
I
norm, then the function
T
dened by
F
T
Gt supTFu, Gs u s t, u, s gt .
is a t
norm on
.
Let T be a continuous t
I
norm. Then the t
norms
T
and T are uniformly continuous
on
,d
L
, where d
L
is a Modied L evy metric see , Denition ...
We nish this section by showing a few properties of the relation dened in Denition .
in the context of t
norms. It T
and T
are continuous t
I
norms, then
T
T
T
T
..
If T is a continuous t
I
norm and T is the t
norm of . then,
iT
T
,
ii M for all t
norms .
Vol. QuasiPseudoMetrization in PqpMSpaces
. Probabilisticquasipseudometric space PqpMspace
Denition .. Let X be a nonempty set, P X
D and T in the t
I
norm. The triple
X, P, T is called a quasipseudoMenger space if it satises the following axioms
MP
xx
u
for all x X,
MP
xz
t
t
TP
xy
t
,P
yz
t
for all x, y, z X and t
,t
gt .
If P satises also the additional condition
MP
xy
u
if x y, the X, P, T is quasiMenger space.
Moreover, if P satises the condition of symmetry P
xy
P
yx
, then X, P, T is called a
Mengerspace.
Denition .. Let X, P, T be a probabilistic quasiMenger space PqM and the
function Q X
D be dened by
Q
xy
P
yx
, x, y X.
Then the ordered triple X, Q, T is also PqMspace. The function Q is called a conjugate
Pqpmetric of the P. By X, P, Q, T we denote the structure generated by the Pqpmetric P
on X.
Denition .. By probabilistic quasi pseudometric space PqpMspace, we mean
an ordered triple X, P, , where X is a nonempty set, the operation is a t
norm and
PX
D satises the following conditions for all x, y, z X,
PP
xx
u
,
PP
xz
P
xy
P
y
z.
Generally, P is called a Pqpmetric. If P satises also the additional condition
PP
xy
u
if x y, then X, P, is called a probabilistic quasimetric space PqM
space.
Note that if, moreover, P satises the condition of symmetry
PP
xy
P
yz
, for all x, y X, then X, P, is a probabilistic metric space PMspace.
Denition .. Let X, P, be a PqpMspace and the function Q X
D be dened
by
Q
xy
P
yx
, x, y X.
Then the triple X, Q, x is also a PqpMspace. The function Q is called a conjugate Pqpmetric
of the P. Let PX, T, Q, denote the structure generated by the Pqpmetric P on X.
Denition .. Let X, P, be PqpMspace. For all x X and t gt , a Pneighborhood
of the point x is the set
U
P
x
tyXP
xy
t gt t y X d
L
P
xy,y
lt t. .
Theorem .. Let X, P, be PqpMspace ander a uniformly continuous t
norm .
Then the family U
p
x
ttR
xX
forms a complete system of neighborhoods in X topolog
P
on X.
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
Also, the Pqpmetric Q which is a conjugate of P generate a topology
Q
. Thus the natural
topological structure associated with a Pqpmetric is a bitopolocal space X,
P
,
Q
.
Lemma .. Let X, P, Q, be a structure dened by a Pqpmetric P. Suppose that
and then function F
X
is dened by
F
xy
P
xy
Q
xy
x, y X. .
Then the ordered triple X, F
, is a Probabilistic pseudo Metricspace. If additionally P
satises the condition
P
xy
u or Q
zy
, x, y X x y .
then X, F
, is a probabilsitc metric space.
Remark .. For an arbitrary
norm , we know that M and we have
F
M
x, y F
x, y, x, y X. .
The function F
M
will be called the natural probabilistic pseudometric generated by the Pqp
metric P. It is the greatest among all the probabilistic pseudometrics generated by P.
. Properties of PE PE PEspaces
Denition .. Fletcher and Lindgren, A quasiuniform structure on a nonempty
set X is a lter U in X X satisfying the following conditions
a Each element U U includes the set x, x x X,
b For every U U, there exists V U such that V V U.
The pair X, U is called a quasiiniform space.
If U is a quasiuniform structure in X, then the family U
U
U U is also
a quasiuniform structure in X which is a conjugate of U. A quasiuniform structure U is a
uniform structure if U U
.
Denition .. Fletcher and Lindgren, . A subfamily B of U is called a base of the
quasiuniform structure if, for all U U, there exists a V B such that V U.
A subfamily S U is called a subbase of U if the family of all nite intersections U
U
U
k
, where U
i
S for i , , , k, is a base of U.
Example .. The discrete quasiuniformity on a set X in fact a uniformity It is the
quasiuniformity D induced by the basis consisting of the diagonal set of X X only. Generally,
nite sets are implicitly equipped with the discrete uniformity.
A basis of a quasiuniformity is symmetrical if all its elements are symmetrical. The
quasiuniformity generated by a symmetrical basis is actually a uniformity. If B is a basis of
a quasiuniformity U, the entourages U
for U B form a symmetrical basis of a uniformity
U
which is the smallest uniformity containing U. In particular U
is also generated by the
entourages U
for U U.
Vol. QuasiPseudoMetrization in PqpMSpaces
Let X, U be a quasiuniform space. The intersection
U
of the elements of U forms a
reexive transitive relation on X that is a quasi order. We also denote by U the equivalence
relation on X associated with the quasi order
U
and given by x
U
y if and only if x
U
y
and y
U
x. If U is a uniformity then
U
and
U
coincide.
If X, U and Y, V are quasiuniform spaces, a mapping f X Y is said to be U, V
uniformly continuous or uniformly continuous if there is no ambiguity if, for each entourage
V V, there exists an entourage U U such that
x, y U fx, fy V.
In particular, such a mapping is monotonous if
x
U
y fx
V
fy.
For each x X, let
U
x
Ux U U.
There exists a unique topology on X, called the topology induced by U, for which U
x
is the lter
of neighborhoods of x for each x X. Note that this topology is not necessarily Hausdor if the
pair x, y of elements of X lies in each entourage U, that is, x
U
y, then each neighborhood
of x contains y.
We implicitly assume that the set X X is endowed with the product topology.
Let X, U be a uniform space. A lter F on X is a Cauchy lter if for each entourage
U U there exists F F such that F F V The uniform space
X, U
is said to be complete
if each Cauchy lter on X is convergent. The Hausdor completion
X,
U
of a uniform space
X, U and the uniformly continuous mapping on X are uniquely dened up to isomorphism
by the following universal property every uniformly continuous mapping f from X, U into
a Hausdor complete uniform space Y, U induces a unique uniformly continuous mapping
f
X Y such that
fx fx for all x X.
Let us now consider a quasiuniform space X, U . We dene the Hausdor completion
X,
U
of X, U to be the Hausdor completion of the uniform space X, U
.
Denition .. A pair X, E is called a probabilistic Espace shortly, PEspace if
X is a nonempty set and E X
is a function which satises the following conditions
for all x, y, z X,
Ex, x u
,.
For each t gt , there exists k gt such that, if E
xy
k gt k .
and E
xz
k gt k, then E
xz
t gt t.
Theorem .. Let X, E be a PEspace. Then the family B Ut t gt , where
Ut is dened by
Ut x, y X
E
xy
t gt t, .
is a base of a quasiuniform structure on X.
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
Proof. For every t gt , by ., the diagonal is a subset of Ut. Next, let lt t
lt t
and let x, y Ut
. By . and the fact that E
xy
is nondecreasing, we infer that
E
xy
t
E
xy
t
gt t
.
Thus we have x, y Ut
, which implies that Ut
Ut
.
Finally, for all t
,t
gt , we obtain
Umint
,t
Ut
Ut
.
It follows directly from . that, for each t gt , there exists k gt such that
Uk Uk Ut.
Thus the condition b of Denition . holds true. This completes the proof.
Corollary .. If U is a quasiuniform structure with a base B dened in Theorem .,
then the family
B
U
n
nN
,
where U
n
is dened by ., is a countable base of U.
Proof. It suces to observe that, for every t gt , there exists n N such that
U
n
Ut.
Indeed, for every t gt , there exists a natural number n such that
t
n
.
Since the functions E
xy
are nondecreasing, we have
E
xy
tE
xy
n
gt
n
gt t.
This means that, if x, y U
n
, then it also belongs to Ut, which shows that
U
n
Ut, t gt .
This completes the proof.
Lemma .. Let X, E be a PEspace and let E
X
be dened by
E
x, y Ey, xx, y X.
Then X, E
is also a PEspace that generates a quasiuniform structure U
which is a
conjugate of U. A base of this structure is a countable family B
consisting of the sets of the
form U
n
, where U
n
B
.
Lemma .. Let X, E be a PEspace and let E E
X
be given by
EE
x, y minEx, y, E
x, y, x, y X.
Vol. QuasiPseudoMetrization in PqpMSpaces
Then the pair X, E E
is a PEspace satisfying the symmetry condition, i.e., for all
x, y X, E E
x, y E E
y, x. Thus E E
generates a uniform structure on X
for which the family B B
is a countable base.
Remark .. Note that, in the proofs of Theorem . and Lemma ., we have used the
monotonicity of E
xy
. However, we have not utilized the leftcontinuity of these functions. We
thus conclude that the codomain of E of Denition . can be extended to the family
of all
nondecreasing functions F , , such that F and F . The obtained
function E X
also generates some quasiuniform structure on X.
Theorem .. Let U be a quasiuniform structure dened by a PEspace X, E.
For all x X and U U, let Ux y X x, y U. Then the family
T
E
G X for each x G, there exists U U such that Ux G
forms a topology on X. The family
B
T
G X for each x G, there is Ut B such that Utx G
is a base of the topology T
E
.
Proof. By the denition of T
E
, it follows that a union of an arbitrary subfamily of T
E
belongs to T
E
. If G
,G
T
E
and x G
G
, then there are U
,U
U such that
U
xG
,U
xG
.
By Denition ., it follows that
UU
U
U.
Since Ux U
U
xG
G
, we have
G
G
T
E
,
which implies that T
E
is a topology on X.
The second part of the proof immediately follows from the denition of a base of a topo
logical space and from Denition .. This completes the proof.
Theorem .. Every quasiuniform structure U generated from a space X, E is quasi
pseudometrizable. This means that there exists a quasipseudometric p which generates a
quasiuniform structure U.
Proof. By Corollary ., U has a countable base B
. The assertion now follows directly
from a theorem by , Theorem .
Remark .. If p is a quasipseudometric which generates a quasiuniform structure U,
then the conjugate structure U
generates a quasipseudometric q which is a conjugate of p.
The uniform structure U U
of Lemma . generates a pseudometric p q maxp, q. It
follows that, if U U
, then the generating function is a pseudometric. In PEspaces, this
holds if and only if E satises the symmetry condition.
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
. Relationships between PE and PqpMspaces
Relationships between probabilistic quasipseudometric spaces and probabilistic Espaces
are given by the following theorems.
Lemma .. Let X, P, T be a quasipseudoMenger space. Let the t
I
norm T satisfy
the condition
supTx, x x lt . .
Then X, P is a PEspace.
Proof. Let t gt . Then, by ., there is k lt
t
such that
T k, k gt t.
By P, we get
P
xz
tT
P
xy
t
,P
yz
t
TP
xy
k, P
yz
k
T k, k
gt t.
This means that .. holds true. Hence X, P is a PEspace. This completes the proof.
Lemma .. Let X, P, be a PqpMspace. Assume additionally that the t
norm
is continuous at the point u
,u
. Then X, P is a PEspace.
Proof. It suces to verify the triangle condition. Let t gt . Then there exists k gt such
that
d
L
G
G
,u
lt t
whenever
d
L
G
,u
lt k
and
d
L
G
,u
lt k.
Then, by P and ., we infer that
d
L
P
xy
P
yz
,u
lt t.
This completes the proof.
An immediate consequence of Theorem . and Lemma . as well as of the Remark .,
we have the following
Theorem .. Let X, P, be a PqpMspace with t
norm being continuous at
u
,u
. Then the bitopological space X, T
U
,T
U
generated from a quasiuniform
structure U is quasipseudometrizable. The topological space X, T
UU
generated from the
uniform structure U U
is pseudometrizable. This means that, if a quasipseudometric p
Vol. QuasiPseudoMetrization in PqpMSpaces
induces the topology T
U
, then the quasipseudometric q, which is a conjugate of p, induces
T
U
.
Lemma .. If X, P, T is a quasipseudoMenger space, then the bitopological space
X, T
U
,T
U
is quasipseudometrizable whenever the t
I
norm T satises the condition ..
Remark .. Note that . and . yield the equality
N
P
x
t Utx, t gt .
We thus infer that the bitopological spaces X, T
P
,T
Q
and X, T
U
,T
U
dened, respectively,
in Theorem . and Lemma . are identical and so we can replace the continuity assumption
about the t
norm by the continuity assumption at the point u
,u
.
On the other hand, as demonstrated by Mustari and
Serstnev , Counterexample for
probabilistic metric spaces, if the t
norm is not continuous at u
,u
, then the family of
sets dened in . does not determine a topology on X, i.e., the following condition fails
For each y N
P
x
t, there exists N
P
y
t
such that
N
P
y
t
N
P
x
t.
Denition .. For a probabilistic quasipseudometric space X, P and s, t gt , we
dene a subset of X
by
U
P
s, t x, y X
P
xy
s gt t. .
Theorem .. Let X, P be a probabilistic quasipseudometric space. Then the quasi
t
I
norm T
PQ
MT
P
,T
Q
, where T
P
I
I is dened by
T
P
a, b infP
xz
t
t
P
xy
t
a, P
yz
t
b
and for all t
I
norms T T
PQ
has the property
supT
PQ
t, t t lt
if and only if, for each t gt , there is t
gt such that, for all s
,s
gt ,
U
P
s
,t
U
P
s
,t
U
P
s
s
, t. .
Proof. For arbitrary t gt , we select t
gt such that
T
PQ
t
,t
gt t.
Next, suppose that P
xy
s gt t
and P
yz
s
gt t
. Then, for T
PQ
, we obtain
P
xz
s
s
T
PQ
P
xy
s
,P
yz
s
T
PQ
t
,t
gt t.
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
On the other hand, for any t gt , by ., choose t
gt . Then, for each t
gt k gt , let
P
xy
s
gt k and P
yz
s
gt k. Then we have
P
xz
s
s
gt t.
Hence, by Lemma ., we have
T
PQ
k, k gt t.
This completes the proof.
Theorem .. If X, P is a statistical quasipseudometric space, then the family U
U
P
s, t s, t gt is a base of a quasiuniform structure on X if and only if, for each pair
s, t, there exists a pair s
,t
such that
U
P
s
,t
U
P
s
,t
U
P
s, t. .
The fact that the quasiuniform structure generated by U has a countable base yields the
following
Corollary .. A quasiuniform space X, U generated by a statistical quasipseudo
metrizable if and only if the condition . holds.
Remark .. Let X, P be a statistical quasipseudometric space. Then the quasi
uniform structures generated by bases dened in . and . are equivalent.
Indeed, we have
U
P
t, t Ut, Umins, t U
P
s, t, s, t gt .
Remark .. The following example shows that the condition . is essentially weaker
than the condition .. Let X , and P X
be given by
P
xy
u
, if x and x y,
u
xy
, otherwise.
Let s
and t gt . Then, for x
and z , we have
P
xy
,u
,
and
P
yz
,u
,.
However, we have
P
xz
u
,.
Thus . does not hold.
Now let s gt and t gt . It suces to select numbers s
gt s
and t
t in order for .
to hold. Thus X, P is a PEspace by Remark .. Notice also that X, P fails to be a
statistical quasipseudometric space.
Vol. QuasiPseudoMetrization in PqpMSpaces
. Pseudometrization in PqpMspaces
The problem of pseudometrization of probabilistic quasipseudometric spaces is charac
terized by the following result, which is a consequence of Theorem ..
Theorem .. Let X, P, be a PqpMspace such that P satises the symmetry condi
tion and the t
norm is continuous at u
,u
. Then the topology T
p
is pseudometrizable.
The topology T
P
is metrizable if the function P satises the condition P of Denition ..
Proof. If P is symmetric, then we have the quasiuniform structure U U
and U
UU
. The assertion is an immediate consequence of Theorem .. For the proof of the
second part of the theorem, notice that it follows from Denition . that x y if and only if
P
xy
u
. This means that
tgt
Ut .
Indeed, let x y. Then we have
d
L
P
xy
,u
t
gt ,
which implies that x, y Ut
and hence x, y does not belong to
tgt
Ut. It follows that
U is a Hausdor uniform structure. Thus the topology generated by it is a Hausdor topology.
This completes the proof.
Remark .. A metrization theorem for Menger spaces was proved by Schweizer and
Sklar , Theorem . Such a theorem for probabilistic metric spaces with a continuous t
I
norm
goes back to Schweizer and Sklar , Theorem .., p. .
Theorem .. Let X, P, be a PqpMspace such that the t
norm is continuous at
u
,u
. If the topology T
P
is stronger than T
Q
, then X, T
P
is pseudometrizable.
Proof. Let T
PQ
be the topology generated by the probabilistic pseudometric F
PQ
.
Topology T
PQ
is the smallest topology containing T
P
and T
Q
. Thus we have
T
P
T
PQ
.
The assertion now follows immediately by the Denition .. This completes the proof.
Theorem .. Let X, P, be a PqpMspace such that the t
norm is continuous at
u
,u
. If P satises the condition
If P
xA
u
, then Q
xA
u
for any x X and A X, .
then X, T
P
is pseudometrizable.
Proof. Let A be Qclosed and let z A
P
. Then we have P
xA
u
. By ., it follows
that also Q
xA
u
, which means that x A
Q
A. We infer that each Qclosed set A is P
closed. Thus T
P
is stronger than T
Q
. The assertion follows from Theorem .. This completes
the proof.
Theorem .. Let X, P, be a PqpMspace such that the t
norm is continuous
at u
,u
. If g
x
X , dened by g
x
yd
L
P
xy
,u
is Qcontinuous, then the space
X, T
is pseudometrizable.
Yeol Je Cho, Mariusz T. Grabiec and Reza Saadati No.
Proof. If g
x
is Qcontinuous, then N
P
x
tT
Q
. Then T
Q
is stronger than T
P
by applying
Theorem ..
Theorem .. Let X, P, be a PqpMspace such that the t
norm continuous. Then
X, T
P
is metrizable provided that X, T
Q
is compact.
Proof. Let G be a Qopen set and take y G. The bitopological space X, T
P
,T
Q
generated by the probabilistic quasimetric P is a pairwise Hausdor space see . This
means that, for every x X G, there exist a Qopen set U and a Popen set V such that
x U, y V, U V .
Observe that the family U x XG is a Qopen cover of xG. By our assumption, there
exists a nite subcover U
,,U
n
. Let V
V
i
i , , n. Then we have y V G.
Thus every Qopen set is also Popen, i.e., T
P
is stronger than T
Q
. The assertion now follows
from Denition .. This completes the proof.
Acknowledgments
The authors would like to thank the Editorial Oce and Y.Y. Liu for their valuable
comments and suggestions.
References
Fletcher, P. and Lindgren, W. F., QuasiUniform Spaces, New York and Basel M.
Dekker, .
Grabiec, M., Probabilistic quasipseudometricspaces, Busefal, , .
Kelly, J.C. Bitopological spaces, Proc. London Math. Soc., , .
Menger, K., Statistical metrics, Proc. Nat. Acad. Sci., USA, , .
Mustari, D.H. and
Serstnev, A.N., A problem about the triangle inequalities for random
normed spaces, Kazan Gos. Univ. Ucen. Zap., , .
Schweizer, B. and Sklar, A., Triangle inequalities in a class of statistical metric spaces,
J. London Math. Soc., , .
Schweizer, B. and Sklar, A., Probabilistic Metric Spaces, ElsevierNorth Holand, .
Reilly, I.L. Subrahmanyam, P.V. and Vamanaurthy, M., On bitopological separation
properties, Nanta Math., , .
Scientia Magna
Vol. , No. ,
On Smarandache least common multiple ratio
S.M. Khairnar
, Anant W. Vyawahare
and J.N.Salunke
Department of Mathematics, Maharashtra Academy of Engg., Alandi, Pune, India.
, Gajanan Nagar, Wardha Road, Nagpur, India.
Department of Mathematics, North Maharashtra University, Jalgoan, India.
Email smkhairnargmail.com vvishweshdataone.in.
Abstract Smarandache LCM function and LCM ratio are already dened in . This paper
gives some additional properties and obtains interesting results regarding the gurate
numbers.
In addition, the various sequaences thus obtained are also discussed with graphs and their
interpretations.
Keywords Smarandache LCM Function, Smarandache LCM ratio.
. Introduction
Denition .. Smarandache LCM Function is dened as SLn k, where SL N
N
n divides the least common multiple of , , , , k,
k is minimum.
Denition .. The Least Common Multilpe of , , , , k is denoted by , , , , k,
for example SL , SL , SL , SL , SL , SL
, SL , .
Denition .. Smarandache LCM ratio is dened as
SLn, r
n, n , n , , n r
,,,,r
.
Example.
SLn, n,
SLn,
n.n
,n,
SLn,
nn n
, if n is odd, n
nn n
, if n is even, n
Proof. Here we use two results
. Product of LCM and GCD of two numbers Product of these two numbers,
. , , , , n , , , , p, p , p , p , , n.
S.M. Khairnar, Anant W. Vyawahare and J.N.Salunke No.
Now,
SLn,
n, n , n
,,
,
Here, n, n , n
n,
nn
n,n
.
But n , n GCD of n and n, which is always .
Hence, n, n , n n, n n and clearly , , .
At n SL,
,,
,,
.
At n SL,
,,
,,,
.
Hence, SLn,
nn n
, if n is odd
nn n
, if n is even
is proved.
Similarly SLn,
n, n , n , n
,,,
, for n
n.n .n .n
, if does not divides n
n.n .n .n
, if divides n
Similarly, SLn,
n.n .n .n .n
, with other conditions also.
Here, we have used only the general valuesof LCM ratios given in and .
The other results can be obtained similarly.
. Sets of SLn, r
SLn, , , , , , , , n, It is a set of natural numbers.
SLn, , , , , ,
nn
, It is a set of triangular numbers.
SLn, , , , , , , , ,
nn n
,.
This set, with more elements, is , , , , , , , , , , , , ,
,,,,,,,,,,,,,,,,,
,,,,,,,,,,,,,.
Its generating function is
x
x
x
x
x
.
Vol. On Smarandache least common multiple ratio
Graph of SLn,
Physical Interpretation of Graph of SLn, This graph, given on the next page, represents the
VI characteristic of two diodes in forward bias mode. It is represented by the equation
II
exp
eV
kBT
, a rectier equation, where,
I
total saturation current,
e charge on electron,
V applied voltage,
kB Boltzmans constant, and
T temperature.
Here V is positive. Xaxis represents voltage V and Yaxis is current in mA.
Also, this graph represents harmonic oscillator Kinetic energy along Y axis and velocity along
Xaxis.
S.M. Khairnar, Anant W. Vyawahare and J.N.Salunke No.
Graph of SLn,
SLn, , , , , , , ,
nn n n
, . . .,
This set, to certain terms is , , , , , , , , , , , ,
,,,,,,,,,,,,,,,
,,,,,,,,,,,,,.
Physical Interpretation of Graph of SLn, This graph, the image of graph about a line
of symmetry y x, is a temperatureresistance characteristic of a thermister.
Its equation is R R
. exp
T
T
, where
R
resistance of room temperature,
R resistance at dierent temperature,
constant,
T
room temperature.
Vol. On Smarandache least common multiple ratio
Graph of SLn,
Temperature T, in Kelvin units, along Xaxis and resistance R, in ohms, along Yaxis,
value lies between and .
The above equation can be put as R C.e
T
.
Its another representation is potential energy in ergs, along Yaxis of system of spring
against extension in cetimeters along Xaxis for dierent weights.
The second graph below is characteristic curve of V
CE
against I
CE
at constant base current
I
B
.
SLn, , , , , , , , ,
nn n n n
, This set, to
certain terms, is , , , , , , , , , , , , ,
Physical Interpretation of Graph of SLn, The second graph of SLn, , given above,
represents the VI characteristic of two diodes in reverse bias mode. It is represented by the
same equation mentioned in graph of SLn, with a change that V is negative.
Hence, V
kBT
e
, and that exp
eV
kBT
, so that I I
.
This shows that the current is in reverse bias and remains constant at I
, the saturation
current, until the junction breaks down. Axes parameters are as above.
Similarly for the other sequences.
S.M. Khairnar, Anant W. Vyawahare and J.N.Salunke No.
. Properties
Murthy formed an interserting triangle of the above sequences by writing them verti
cally, as follows
. Here, the rst column and the leading diogonal contains all unity.
The second column contains the elements of sequence SLn, .
The third column contains the elements of sequence SLn, .
The fourth column contains the elements of sequence SLn, .
and similarly for other columns.
. Consider that row which contains the elements only as rst row.
If p is prime, the sum of all elements of p
th
row mod p.
If p is not prime, the sum of all elements of
th
row mod .
The sum of all elements of
th
row mod .
The sum of all elements of
th
row mod .
The sum of all elements of
th
row mod .
The sum of all elements of
th
row mod .
. Dierence
We have,
SLn, SLn , SLn , .
This needs no verication.
Also, SLn, SLn ,
nn n
nnn
nn
SLn , .
Vol. On Smarandache least common multiple ratio
Similarly,
SLn, SLn , SLn ,
SLn, SLn , SLn , .
Hence, in general,
SLn, r SLn , r SLn , r , r lt n.
. Summation
Adding the above results, we get,
r
SLn, r n , n gt .
. Ratio
We have,
SLn,
SLn,
n
,
SLn,
SLn,
n
,
SLn,
SLn,
n
.
In general,
SLn, r
SLn, r
nr
r
.
. Sum of reciprocals of two cosecutive LCM ratios
We have,
SLn,
SLn,
n
SLn,
,
SLn,
SLn,
n
SLn,
,
SLn,
SLn,
n
SLn,
.
In general,
SLn, r
SLn, r
n
r SLn, r
.
S.M. Khairnar, Anant W. Vyawahare and J.N.Salunke No.
. Product of two cosecutive LCM ratios
. SLn, SLn,
n
n
. SLn, SLn,
n
n
n
. SLn, SLn,
n
n
n
n
. SLn, SLn,
n
n
n
n
n
In general,
SLn, r SLn, r
n
n
n
n
nr
nr
rr
.
References
Amarnath Murthy, Some notions on Least common multiples, Smarandache Notions
Journal, , .
Maohua Le, Two Formulas of Smarandache LCM Ratio Sequences, Smarandache No
tions Journal, , .
Wang Ting, Two Formulas of Smarandache LCM Ratio Sequences, Scientia Magna,
, No., .
Scientia Magna
Vol. , No. ,
Some New Relative Character Graphs
R. Stella Maragatham
Department of Mathematics, Queen Marys College, Chennai, India
Email r stellaeyahoo.co.in
Abstract In this paper based on the group characters there are certain graphs constructed.
The properties are analysed with examples.
Keywords IrrG, RCgraphs, Core
G
H
. Introduction
This paper is the outcome of the author in throwing some more light on the relative
character Graphs G, H RCgraphs originally studied by T. Gnanaseelan in his Ph.D work
.
It may be recalled that in the early s R. Brauer while studying the ordinary complex
irreducible representation and the modular irreducible representation of a nite group G,
constructed a nite simplex graph which was later called the Brauer Graph BG and was
extensively studied especially when G is a nite Chevalley group, or more generally, a simple
group of his type .
Both these graphs have the same vertex set namely, the set IrrG of complex irreducible
Characters of G. In the Brauer group case, case vertices , are adjacent if and only if their
reduction modulo a prime dividing OG contains at least one modular irreducible character
in common. In the case of the RCgraph and are meant to be adjacent if and only if the
restriction
H
and
H
to a given subgroup H of G contain at least one irreducible character
of H in common. Clearly G, H is a simple graph.
We shall highlight some of the basic properties obtained by Gnanaseelan and others. This
paper is a continuation of the earlier works deals with connectivity properties of the
complement
G, H.
. Basic properties of RCgraphs
We need the following base minimum from character theory. For details, we refer to M.
Isaacs book .
Given two characters , of G, let , /Gss. Then, Q is irreducible if
and only if , and if and are two distinct irreducible, then , .
R. Stella Maragatham No.
In terms of this scalar product the adjacent condition can be given as follows
, IrrG are adjacent if and only if ,
H
OH
sH
ss gt
Induced characters Given any character Q of a subgroup H, the induced character Q
G
of
G is dened as
Q
G
s
OH
g GQ
gsg
, where
Q
gsg
Qgsg
if gsg
H and
, otherwise.
Frobenius Reciprocity Formula
If Q IrrH and IrrG, then
H
,Q,Q
G
G
.
Proposition . Two elements , of IrrG are adjacent i
where
G
H
.
For details and proofs of the following results, we refer to , and .
Proposition . G, H is connected if and only if core
G
H where core
G
H is the
largest normal subgroup of G
G
contained in H. In particular if G is simple or if H is simple,
then G, H is connected.
For the above proposition and for further results, we need the path lemma Given any F
in the connected component containing
G
, in connected to
G
by a path of length s if and
only if
s
,s.
Proposition . G, H is a tree if and only if G NH is a Frobenius group with kernel
N and complement H and N is unique minimal elementary abelian of order p
m
, aprime and
Hp
m
. In this case, the tree is always a star.
Proposition . , IrrG lie in the same connected components of G, H if and only
if
s
for some integer s .
Proposition . If G, H is connected then it is always triangulated . A tree is trivially
triangulated
. Complements of RCgraphs
If H is a subgroup of G, it is very rare that the complements G, H of G, H is of the
form G, H for some subgroup K.
Problem . Characterize all groups G with the property that there exists a pair of
subgroups H and K such that
G, H G, H.
In this connection the following results of Gnanaseelan is interesting proposition If g NH
is a semi direct product with N normal and H nonnormal, then G, N G, H if and
only if G is Frobenius with kernel N and complement H.
We shall now focus on the connectivity of G, H and G, H also see .
At the outset, if a graph is disconnected then is connected. However, if is connected,
then may be connected or not, as seen from the following examples
Vol. Some New Relative Character Graphs
GS,HH.
G, H
G
G, H G, H
G, H
G S , H s.
Problem . Given any nite group G, nd all subgroups H such that G, H is connected.
we can assume core
G
H.
Problem . Find all nite groups G such that whenever H , then G, H is
connected.
Note that this includes the class of all nite abelian groups, because, if G is abelian of order
g and H is a subgroup of order h, then G, H is disconnected with exactly h components
each component being complete with g/h vertices.
Theorem . Let H be a subgroup of G such that the right action of G on G/H is doubly
transitive. If G, H is not a tree and if q lt n C
, then G, H is connected where n
and q are the number of vertices and number of edges of G, H respectively.
Proof of Theorem . If G, H is disconnected then G, H is always connected.
Hence we assume that G, H is connected and is of the form
G
Since the action of G on G/H is doubly Transitive,
G
H
G
, IrrG.
Let J denote the sub graph G, H
G
. Then J has n vertices and q edges. By
Frobenuis reciprocity formula we have,
H
H
r
i
Q
i
,Q
i
IrrH, r
i
gt
We shall prove that there is at least one IrrG,
G
which is not adjacent to .
For any Q IrrH, let IQ denote the set of distinct irreducible characters occurring in Q
G
.
R. Stella Maragatham No.
First suppose that there exists Q IrrH such that IQ J and
H
is irreducible for
every IQ. In other words,
H
Q for every IQ. Then clearly any IQ
is such that
H
does not contain Q and hence IQ is a connected component of G, H,
contradicting our assumption that G, H, is connected.
Next, we shall prove that if J is not Irreducible, then . In fact,
,
x
OG
sG
sss
OG
k
i
sC
i
sss
Where C
,C
,...,C
k
are the conjugacy classes of G
OG
k
i
sssC
i
Where s runs through a set of class representatives
OG
k
i
ssC
i
H
using the relation C
i
H Cis
OG
sH
ss
OG
OH,
H
d gt
since
H
is not irreducible using
G
, this gives , gt , which means .
Already by assumption and are adjacent, which is by path lemma, gives
. Using
these relations, we get
. Hence by path lemma, we get a path of length from
to
G
, which must pass through . Hence we get a cycle for some J. Repeating
this process over and over again, we see that is adjacent to any in J.
Now Suppose, for any Q IrrH, there exists IQ such that
H
is not irreducible.
Then from the above argument, it follows that is adjacent to any in J. Then J is a complete
graph and hence q n C
including the edge
G
which is a contradiction to our
assumption that q n C
.
Hence there must exist a vertex not adjacent to . In the complement, therefore, and
are adjacent. Also, since is the only vertex adjacent to
G
in G, H, all other vertices
including are adjacent to
G
in G, H. Thus G, H is connected.
Remarks .
. The assumption that q n C
is necessary as we see from the following graph. Take
GA
and H A
.
Vol. Some New Relative Character Graphs
G
G, H
Here n and q , which is gt C
of course G, H is disconnected.
. The condition that G, H is not a tree is necessary, for otherwise, G, H would be a
rooted tree star and hence would be isolated in the complement.
Problem . Characterize all subgroups H of a given group G such that both G, H and
G, H are connected.
Theorem . G, H is connected if and only if for every I
H
there exists a
V
H
such that and are not adjacent in G, H.
Proof of Theorem . First assume that G, H is connected. Let I
H
. Suppose
is adjacent to all V I
H
in G, H. Since already is adjacent to all I
H
,
as I
H
is complete, is adjacent to all vertices in G, H. Hence is an isolated vertex in
G, H, contradiction.
For the converse part, assume the given condition. Since
G
is not adjacent to any vertex
in V I
H
,
G
is adjacent to every vertex in V I
H
in G, H. By the assumption, if
I
H
and V I
H
, and are adjacent in G, H . . . . . . . . . . . . . . . . . . . . . . . . . .
Also
G
is adjacent to all vertices in V I
H
in G, H. In fact, no vertex can be
adjacent to
G
without belonging to I
H
. Therefore, if V I
H
, then is not adjacent
to
G
in G, H. Hence is adjacent to
G
in G, H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Now take any two , in .
Case Both and belong to I
H
. By , there exist
,
in V I
H
, such that is
adjacent to
and is adjacent to
in G, H. But by ,
G
is adjacent to both
and
in G, H. Then we get a path between and . see gure .
G
Figure
Case Let I
H
and V I
H
, By ,
G
is adjacent to in G, H, and
by , their exists V I
H
such that is adjacent to in G, H. This shows that
there is a path between and in G, H. see gure .
Case Both and do not lie in I
H
. By ,
G
is adjacent to and in G, H
hence there is a path between and in G, H. see gure .
R. Stella Maragatham No.
G
Figure
G
Figure
The above arguments prove that G, H is connected. Hence the theorem.
Theorem . Let the right action of G and G/H be doubly transitive. Then G, H is
connected if and only if diamG, H .
Proof of Theorem . Due to doubly transitivity, I
H
G
, IrrG, and hence is
the unique vertex adjacent to
G
. First let diamG, H . Then their exists V I
H
whose distance from is at least . Therefore, given I
H
we can nd V I
H
such that and are not adjacent. Hence by theorem , G, H is connected.
Conversely, assume that G, H is connected. Hence by the same theorem, their exists
VI
H
such that and are not adjacent. Therefore, dist., . Hence
diamG, H . This proves the theorem.
Proposition .
i If G NH is Frobenius, then rG, H . As an example D
,C
is the following.
G
Of course there are cases when rG, H gt . For instance, when G S
and H S
,
G, H is the graph.
G
Whose domination number is . For details on domination theory of RCgraphs, we refer
Vol. Some New Relative Character Graphs
to , .
ii RCgraphs as signed graphs. There is a natural sign attached to the edges. If , are
the vertices and
H
m
i
G
i
and
H
n
i
Q
i
, so that ,
H
m
i
n
i
, the edge
gets a sign if m
i
n
i
is even and a sign if m
i
n
i
is odd. The rst author has initiated
a study on RCgraphs as signed graphs following the work of B.D. Acharya et al. see
.
iii Among the many products occurring in graph theory the one that suits our RCgraph is
the strong direct product. The domination number of a strong direct product cannot be
greater than that of the more fancied Cartesian product
If
and
are two connected graphs then we have the famous vizings conjecture
. A variation of the above conjunctive for RCgraphs and strong
direct products fails. For details see .
Conclusion
Graph theorists have already shown interest in RCgraphs as they provide some relief
from the monotonous edge dot study The authors believe that the pictorial description of
character theory could help people to understand representation theory in a better way. The
restriction behavior of an irreducible G character to a subgroup H, apart from Cliords
theory when H is normal, is not completely understood. Last but not the least many new
perhaps hitherto unseen, subgroups have already been located in an arbitrary nites groups
using RCgraphs such as almost normal subgroups, dominate subgroups, domination free
subgroups etc, whose denition and details are omitted here for want of time and space.
References
R. Brauer, On the connection between the ordinary and modular characters of groups
of nite order, Ann. of Math., , .
T. Gnanaseelan, Studies in Group Representation, Ph. D. Thesis, Madurai Kamaraj
University .
M. Isaacs, Character Theory of Finite Groups, Academic Press, .
A.V. Jeyakumar, Construction of some new nite graphs using group representations,
Proceedings of the Conference on Graphs, Combinatorics, Algorithms and Applications A.
Kalasalingam College, Narosa Publication, .
Mohammed Sheri, Ph.D. Thesis, Madurai Kamaraj University, .
R. Stella Maragatham, Studies in Group Algebras and Representations, Ph. D. Thesis,
Madurai Kamaraj University, .
Scientia Magna
Vol. , No. ,
Class A weighted composition operators
D. Senthilkumar
and K. Thirugnanasambandam
Department of Mathematics, Government Arts College Autonomous, Coimbatore .
Department of Mathematics, Sri Ramakrishna Engineering College, Coimbatore ,
India.
Email senthilsenkumharigmail.com kthirugnanasambandamgmail.com
Abstract In this paper Class A weighted composition operators on L
spaces are charac
terized and their various properties are studied.
Keywords Composition operators, class A operators.
. Preliminaries
Let X, , be a sigmanite measure space and let T X X be a nonsingular
measurable transformation. Let L
L
X, , . Then the composition transformation C
T
is
dened by C
T
f f T for every f in L
. If C
T
happens to be a bounded operator on L
,
then we call it the composition operator induced by T.
C
T
is a bounded linear operator on L
precisely when i the measure T
is abso
lutely continuous with respect to and ii the RadonNikodym derivative dT
/d is in
L
X, , . Let RC
T
denote the range of C
T
and C
T
, the adjoint of C
T
.
A weighted composition operator is a linear transformation acting on a set of complex
valued mesurable functions of the form Wf wf T, where w is a complex valued
measurable function. In case w a.e., W becomes a composition operator, denoted by C
T
.
To examine the weighted composition operators eciently, Lambert , associated with
each transformation T, the so called conditional expectation operator ET
E. More
generally, Ef may be dened for bounded measurable function f or nonnegative measurable
functions f for details on the properties of E, see , , .
As an operator on L
P
, E is the projection onto the closure of the range of C
T
. E is the
identity on L
P
if and only if T
.
The RadonNikodym derivative of T
with respect to is denoted by h and that of
T
k
with respect to is denoted by h
k
where T
k
is obtained by composing T with itself k
times. Let w
k
denote ww T wT
wT
k
so that W
k
fw
k
fT
k
.
. Class A composition operators
Let BH denote the Banach algebra of all bounded linear operators on a Hilbert space H.
An operator T BH is said to be hyponormal if T
T TT
, T is said to be phyponormal
Vol. Class A weighted composition operators
if T
T
p
TT
p
, lt p lt . For a phyponormal operator T UT Aluthge introduced
the operator
TT
/
UT
/
which is called Aluthge transformation and Aluthge showed
very interesting results on T. The operator T is said to be w hyponormal if
TT
T
. T is paranormal if T
x x Tx
T is quasihyponormal if T
TT
x for all
x in H or equivalently T
T
T
T
.
An operator T belongs to class A if T
T
. Furuta, Masatoshi Ito and Takeaki
Yamazaki have characterized class A operators as follows. An operator T belongs to class A
if
and only if T
T
T
/
T
T.
. Previous results on Mparanormal composition opera
tors
Panayappan and Veluchamy have characterized Mparanormal composition operators
as follows.
Theorem .. Let C
T
BL
. Then C
T
is Mparanormal if and only if M
h
kf
k
a.e., for all k R, where f
is the RadonNikodym derivative of T
with respect
to and h
is the RadonNikodym derivative of TT
with respect to .
Also Panayappan generalized the above result to the weighted composition operators
as follows.
Theorem .. W is Mparanormal if and only if M
h
Ew
T
h
Ew
T
a.e..
The aim of the paper is to characterize class A weighted composition operators and also
to show that class A and paranormal operators coincide in the case of weighted composition
operators.
. Weighted class A composition operators
Theorem .. W is of class A if and only if h
Ew
T
h
Ew
T
a.e..
Proof. We have, W
k
fw
k
fT
k
and W
fh
k
Ew
k
fT
k
and so W
W
fW
w
fT
h
Ew
fT
T
h
Ew
T
f.
Also W
Wf hEw
T
f.
Then W is of class A
W
W
W
/
W
W
that is W
W
WW
W
if and only if W
W
WW W
W
W
W
W
W
W
W
W
W
f, f for every f
E
h
Ew
T
h Ew
T
f
d for every E
h
Ew
T
h
Ew
T
a.e.
D. Senthilkumar and K. Thirugnanasambandam No.
Theorem .. Let W be a weighted composition operator with weight w gt . Then the
following are equivalent
i W is paranormal
ii W is class A
iii W is quasihyponormal.
Proof. i ii
Suppose W is paranormal. Then by Theorem .
h
Ew
T
h
Ew
T
a.e.
W is class A.
ii iii
Suppose W is class A. Then by Theorem ., h
Ew
T
h
Ew
T
a.e..
Therefore by Corollary . , W is quasihyponormal.
iii i
Suppose W is quasihyponormal, then
h
Ew
T
h
Ew
T
a.e.
Therefore by Corollary ., , W is paranormal.
References
Alan lambert, Hyponormal Composition Operators, Bull London Math. Soc., ,
.
A. Aluthge, On phyponormal Operators for lt p lt , Integral Equations Operators
Theory, , .
A. Aluthge and Derming Wang, An Operator Inequality which implies Paranormality,
Mathematical Inequalities amp Applications, , No., .
J. Campbell, J. Jamison, On Some Classes of Weighted Composition Operators, Glasgow
Math. J., , .
M. Embry Wardrop, Alan Lambert, Measurable Transformations and Centered Com
position Operators, Proc. Royal Irish Acad., , No.A, .
S.R. Foguel, Selected Topics in the Study of Markov Operators, Carolina Lectures Series
Dept. Math., UNCCH, Chapel Hill, N.C. , .
T. Furuta, Masatoshi Ito, Takeaki Yamazaki , A Subclass of Paranormal Operators in
cluding Class of Loghyponormal and Several related classes, Scientiae Mathematicae, ,
No., .
S. Panayappan, Nonhyponormal Weighted Composition Operators, Indian J. Pure
Appl. Math., , No., .
T. Veluchamy and S. Panayppan, Paranormal Composition Operators, Indian J. Pure
Appl. Math., , No., .
Scientia Magna
Vol. , No. ,
Position vectors of some special spacelike
curves according to Bishop frame in Minkowski
space E
S uha Ylmaz
Dokuz Eyl ul University, Buca Educational Faculty, Department of Mathematics,
BucaIzmir, TURKEY
Email suha.yilmazyahoo.com
Abstract In this work, we investigate position vectors of some special spacelike curves with
respect to Bishop frame in E
. A system of dierential equation whose solution gives the
components of position vector on the Bishop axis is established. Via its special solutions,
some characterizations are presented.
Keywords Minkowski space, Bishop frame, spacelike curve.
. Introduction and preliminaries
In the existing literature, it can be seen that, most of classical dierential geometry topics
have been extended to Lorentzian manifolds. In this process, generally, researchers used a
standard moving Frenet frame. Some of kinematical models were adapted on this moving
frame, due to tranformation matrix among derivative vectors and frame vectors. Thereafter,
researchers aimed to have an alternative frame for curves and other applications. Bishop
frame,
which is also called alternative or parallel frame of the curves, was introduced by L.R. Bishop
in . And, this frame have been used in many research papers, in classical manner or
Lorentzian manifolds, etc.
In this work, we consider a spacelike curve with a spacelike binormal. Then, with the
notion of Bishop frame, we investigate position vectors of some special spacelike curves in
Minkowski space E
.
To meet the requirements in the next sections, here, the basic elements of the theory of
curves in the space E
are briey presented. A more complete elementary treatment can be
found in and .
The Minkowski space E
is the Euclidean space E
provided with the standard at
metric given by
, dx
dx
dx
,
where x
,x
,x
is a rectangular coordinate system of E
. Since , is an indenite metric,
recall that a vector v E
can have one of three Lorentzian characters it can be spacelike
if v, v gt or v , timelike if v, v lt and null if v, v and v . Similarly, an
S uha Ylmaz No.
arbitrary curve s in E
can locally be spacelike, timelike or null lightlike, if all of its
velocity vectors
are respectively spacelike, timelike or null lightlike, for every s I R.
The pseudonorm of an arbitrary vector a E
is given by a
a, a. is called an unit
speed curve if velocity vector v of satises v . For vectors v, w E
it is said to be
orthogonal if and only if v, w .
Denote by T, N, B the moving Frenet frame along the curve in the space E
. For a
spacelike curve with rst and second curvature, and in the space E
, the following Frenet
formulae are given in .
Let be a spacelike curve with a spacelike binormal, then the Frenet formulae read
T
N
B
T
N
B
,
where
N, N , T, T B, B ,
T, N T, B T, N N, B .
The Bishop frame is due to L.R. Bishop . This frame or parallel transport frame is an
alternative approach to dening a moving frame that is well dened even the spacelike curve
with a spacelike binormal has vanishing second derivative . He used tangent vector and any
convenient arbitrary basis for the remainder of the frame. Then, the Bishop frame is
expressed
as
T
N
N
k
k
k
k
T
N
N
,
where
s
k
k
,s
d
ds
s arg tanh
k
k
.
Here, we shall call k
and k
as Bishop curvatures.
. Main results
Let s be a spacelike curve with a spacelike binormal. We can write position
vector with respect to Bishop frame as
sTN
N
.
Dierentiating both sides of and considering system , we have a system of dierential
equation as follows
k
k
k
k
.
Vol. Position vectors of some special spacelike curves according to Bishop frame
This system of ordinary dierential equations is a characterization for the curve s.
Position vector of a spacelike curve can be determined by means of solution of it. However,
general solution have not been found. Owing to this, we give some special values to
components
and Bishop curvatures.
Case I. . In this case s lies fully in N
N
subspace. Thus, we have other
components
constant c
constant c
.
System and yield the following linear relation among Bishop curvatures
c
k
c
k
.
Since, we immediately arrive the following results.
Theorem . Let s be a spacelike curve with a spacelike binormal and lies fully
in N
N
.
i If one the second and third components of the position vector of on Bishop axis is
zero, then transforms to a line.
ii There is a relation among Bishop curvatures as .
iii Position vector of can be written as
sc
N
c
N
.
Case II. constant . In this case, rst we have by
and constant.
Suce it to say that this case is congruent to case I.
Case II. a. Let us suppose . Thus, and constant, and so k
is constant.
This case yields a line equation as follows
sN
.
Case III. The case constant is also congruent to case I.
Case III. a. . Then, we easily have k
and are constants. This result follows a
line equation as
sN
.
References
B. B ukc u and M. K. Karacan, The Bishop Darboux Rotation Axis of the Spacelike
Curve in Minkowski Space, E. U. F. F., JFS, , .
B. ONeill, SemiRiemannian Geometry, Academic Press, New York, .
L. R. Bishop, There is More Than one way to Frame a Curve, Amer. Math. Monthly,
, No., .
Yucesan, A., Coken, A.C., Ayyildiz, N., On the Darboux Rotation Axis of Lorentz
Space Curve, Appl. Math. Comp., , .
Scientia Magna
Vol. , No. ,
Monotonicity and logarithmic convexity
properties for the gamma function
Chaoping Chen
and Gang Wang
. College of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan
, China
. Department of Basic Courses, Jiaozuo University, Jiaozuo City, Henan , China
Abstract We present the monotonicity and logarithmic convexity properties of the functions
involving the gamma function. MincSathre inequality are extended and rened.
Keywords Gamma function, monotonicity, logarithmically convexity, inequality.
. Introduction and results
The classical gamma function
x
t
x
e
t
dt x gt
is one of the most important functions in analysis and its applications. The logarithmic deriva
tive of the gamma function can be expressed in terms of the series
x
x
x
n
n
xn
x gt . is the Eulers constant, which is known in literature as
psi or digamma function. We conclude from by dierentiation
k
x
k
k
n
xn
k
x gt k , , ,
k
are called polygamma functions.
H. Minc and L. Sathre proved that the inequality
n
n
lt
n
n
n
n
lt
is valid for all natural numbers n. The inequality can be rened and generalized as see ,
,
nk
nmk
lt
nk
ik
i
/n
nmk
ik
i
/nm
nk
nmk
,
The rst author was supported in part by Natural Scientic Research Plan Project of Education
Department
of Henan Province A, and Project of the Plan of Science and Technology of Education
Department
of Henan Province .
Vol. Monotonicity and logarithmic convexity properties for the gamma function
where k is a nonnegative integer, n and m are natural numbers. For n m , the equality
in is valid. The inequality can be written as
nk
nmk
lt
n k /k
/n
n m k /k
/nm
nk
nmk
.
In , D. Kershaw and A. Laforgia showed the function
x
x
is strictly decreasing
and x
x
x
strictly increasing on , , from which the inequalities can be derived.
In , B. N. Guo and F. Qi proved that the function
fx
x y /y
/x
xy
is decreasing in x for xed y , from which the lefthand side inequality of can be
obtained.
In this paper, our Theorem considers the monotonicity and logarithmic convexity of the
function f on , . This extends and generalizes Guo and Qis result.
Theorem . Let s be real number, then the function
fx
x s /s
/x
xs
is strictly decreasing and strictly logarithmically convex on , . Moreover,
lim
x
fx e
s
/s and lim
x
fx e
.
Theorem . Let s be real number, then the function
gx
x s /s
/x
xs
is strictly increasing on , .
The following corollaries are obvious.
Corollary . Let s be a real number, then for all real numbers x gt ,
e
lt
x s /s
/x
xs
lt
e
s
s
.
Both bounds in are best possible.
Corollary . Let gt and s be real numbers, then for all real numbers x gt ,
xs
xs
lt
x s /s
/x
x s /s
/x
lt
xs
xs
.
In particular, taking in x n, s and , we obtain
n
n
lt
n
n
n
n
lt
n
n
.
The inequality is an improvement of .
Chaoping Chen and Gang Wang No.
. Proof of the theorems
Proof of Theorem . Dene for x gt ,
ux x
f
x
fx
ln
xs
s
xx s
x
xs
,
vx x
d
ln fx
dx
ln
xs
s
xx s
x
xs
x
xs
.
Dierentiation of ux gives
x
u
x
xs
xs
s
xs
n
xsn
n
xsn
xsn
n
s
xsn
s
xsn
n
s
xsn
x s nx s n
s
xsn
n
sxsns
xsn
xsn
lt .
Hence, the function ux is strictly decreasing and ux lt u for x gt , which yields the
desired result that f
x lt for x gt .
Using the asymptotic expansion , p.
ln x
x
ln x x ln
x
Ox
x,
we conclude from
ln fx
x
ln x s ln s lnx s
that
lim
x
fx e
.
By L Hospital rule, we conclude from that
lim
x
fx
e
s
s
.
Vol. Monotonicity and logarithmic convexity properties for the gamma function
Dierentiation of vx yields
x
v
x
xs
xs
s
xs
n
xsn
n
xsn
xsn
n
s
xsn
s
xsn
n
sxsn
sxsns
xsn
xsn
gt .
Hence, the function vx is strictly increasing and vx gt v for x gt , which yields the
desired result that
d
ln fx
dx
gt for x gt .
Proof of Theorem . Dene for x gt ,
pxx
g
x
gx
ln
xs
s
xx s
x
xs
.
Dierentiation of p x gives
x
p
x
xs
xs
s
xs
n
xsn
xs
s
xs
gt
dt
xs
xs
s
xs
x
xs
gt .
Hence, the function p x is strictly increasing and p x gt p for x gt , which yields the
desired result that g
x gt for x gt .
References
H. Minc and L. Sathre, Some inequalities involving r
/r
, Proc. Edinburgh Math.
Soc., /, .
B.N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions,
Taiwanese J. Math., , No., .
F. Qi, Inequalities and monotonicity of sequences involving
n
n k/k, Soochow J.
Math., , No., .
F. Qi and Q.M. Luo, Generalization of H. Minc and J. Sathres inequality, Tamkang
J. Math., , No., .
D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad.
Sci. Torino Cl. Sci. Fis. Mat. Natur., , .
M. Abramowitz and I. A. Stegun Eds., Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathe
matics Series , th printing, with corrections, Washington.
Scientia Magna
Vol. , No.,
On ideas and congruences in KUalgebras
Chanwit Prabpayak
and Utsanee Leerawat
Faculty of Science and Technology Rajamangala University of Technology Phra Nakhon,
Bangkok, Thailand
Department of Mathematics Kasetsart University, Bangkok, Thailand
Email chnwt.pgmail.com fsciutlku.ac.th
Abstract In this paper, we will introduce some kind of algebras which is called KUalgebras.
We dene ideals and study congruences on KUalgebras, and also investigate some related
properties.
Keywords ideals, congruences, KUalgebras
. Preliminaries
Several authors ,,,, introduced some structures of algebras such as BCK, BCC
and BCIalgebras. In , C.S. Hoo introduced the notions of lters and commutative ideals
in BCIalgebras. and introduced a concept of ideals in BCC and BCKalgebras. W. A.
Dudek and X. Zhang gave connections between ideals and congruences in BCCalgebras.
The
objective of this paper is to introduce KUalgebras and also study ideals and congruences in
KUalgebras. Moreover, we investigate some of its properties.
By an algebra G G, , we mean a nonempty set G together with a binary operation,
multiplication and a some distinguished element . In the sequel a multiplication will be
denoted
by juxtaposition.
Denition . An algebra G G, , is called a KUalgebra if it satises the following
conditions
xyyzxz
x x,
x,
xy yx implies x y
for all x, y, z G.
By , we get xx . It follows that xx for all x G. And if we put
y in , then we obtain zxz for all x, z G.
Example . Let G , , , and H , , , , . Let the multiplication of G and
H be dened by Table and Table respectively. Then it is easily checked that G and H are
KUalgebras.
Vol. On ideas and congruences in KUalgebras
Table .
Table .
. Ideals
Denition . Let A be a nonempty subset of a KUalgebra G. Then A is said to be an
ideal of G if it satises the following conditions
i A,
ii for all x, y, z G, xyz A and y A imply xz A.
Example . In example , let A , and B , be subsets of a KUalgebra H.
Then A is an ideal of H but B is not an ideal of H.
Putting x in Denition we obtain the following Proposition
Proposition . Let A be an ideal of a KUalgebra G. Then for all x, y G, xy A and
x A imply y A.
Denition . Let G, , be a KUalgebra. Then a nonempty subset S of G is said to
be a KUsubalgebra of G if S, , is KUalgebra.
Note that S is a KUsubalgebra of G if and only if xy S for all x, y S.
Proposition . Let A be an ideal of KUalgebra G. Then A is a KUsubalgebra of G.
Proof of Proposition . Let x, y A. Then xyxy . Hence yxy A.
Since A is an ideal of G and y A, xy A. Therefore, A is a KUalgebra of G.
Proposition . Let G be a KUalgebra and A a nonempty subset of G with . Then A
is an ideal of G if and only if x A, yz / A imply yxz / A for all x, y, z G.
Proof of Proposition . Let A be an ideal of G and let x A, yz / A. Suppose that
yxz A. Since A is an ideal, yz A, a contradiction.
Conversely, assume that x A, yz / A imply yxz / A for all x, y, z G. Let x, y, z G
be such that xyz A and y A. It is clear that xz A.
Chanwit Prabpayak and Utsanee Leerawat No.
Corollary . Let G be a KUalgebra and A a nonempty subset of G with . Then A is
an ideal of G if and only if x A, y / A imply yx / A for all x, y G.
On KUalgebra G, , . We dene a binary relation on G by putting x y if and only if
yx . Then G is a partially ordered set and is its smallest element. Thus a KUalgebra
G satises conditions yzxz xy, x, x y x implies x y.
. Congruences
In this topic, we describe congruences on KUalgebras. We start with the following
Denition . Let A be an ideal of KUalgebra G. Dene the relation on G by
x y i xy A and yx A
Theorem . If A is an ideal of KUalgebra G, then the relation is a congruence on G.
Proof of Theorem . It is clear that this relation is reexive and symmetric. Let
x, y, z G be such that x y and y z. Then xy, yx, yz, zy A and xyyzxz A.
By Proposition , xz A. Similarly, zyyxzx A. Thus x z. Therefore, is an
equivalenec relation.
If x u and y v for x, y, u, v G, then xu, ux, yv, vy A and xuuyxy A.
By Proposition , uyxy A. Similarly xyuy A. Thus xy uy. On the other hand
uyyvuy A and yv A imply uyuv A. Similarly, if uvvyuy A
and vy A we obtain uvuy A. Thus uy uv. Since is transitive, xy uv. Hence
is a congruence.
Proposition . If is a congruence on a KUalgebra G, then C
x G x is an
ideal of G.
Proof of Proposition . Obviously C
. If xyz C
and y C
, then xyz
and y . Since x x and z z, xyz xz. Thus xz . Hence xz C
. So C
is an
ideal of G.
Let be a congruence relation on a KUalgebra G and let C
x
y G y x. Then
the family C
x
x G gives a partition of G which is denoted by G/
. For any x, y G, we
dene C
x
C
y
C
xy
. Since has the substitution property, the operation is welldened. It
is easily checked that G/
,,C
is a KUalgebra.
Denition . Let G, , and H, , be KUalgebras. A homomorphism is a map
f G H satisfying fx y fx fy for all x, y G. An injective homomorphism is
called monomorphism and a surjective homomorpism is called epimorphism.
The kernel of the homomorphism f, denoted by kerf, is the set of elements of G that map
to in H.
If f is a homomorphism from a KUalgebra G into a KUalgebra H, then we dene kernel
kerf f
as in and we can prove the following result
Theorem . Let G be a KUalgebra and A an ideal of G. Let G/
be a KUalgebra
determined by A. Then the canonical mapping f G G dened by fx C
x
is an
epimorphism and kerf is an ideal of G.
Vol. On ideas and congruences in KUalgebras
Proof of Theorem . Let x, y G. If x y, then C
x
C
y
. Thus fx fy. Since
fxy C
xy
C
x
C
y
fxfy, f is a homomorphism. Let C
x
G/
. We get fx C
x
.
That is f is an epimomorphism.
Since f C
, kerf . If xyz kerf and y kerf. Then fxyz fy C
.
Thus C
fx fy fz fx C
fz fx fz fxz. Thus xy kerf. It
follows that kerf is an ideal of G.
Acknowledgements
This work was supported by a grant from the Graduate School, Kasetsart University.
References
W.A. Dudek and X. Zhang, On ideals and congruences in BCCalgebras, Czechoslovak
Math. Journal, , .
C.H. Hoo, Filters and ideals in BCIalgebras, Math. Japon., , .
J. Meng, On ideals in BCKalgebras, Math. Japon., , .
K. Iseki, On BCIalgebras, Math. Seminar Notes, , .
K. Iseki, and S. tanaka, Ideal theory of BCKalgebras, Math. Japon., , .
Scientia Magna
Vol. , No. ,
Browders theorem and generalized Weyls
theorem
Junhong Tian and Wansheng He
School of Mathematics and statistics Tianshui Normal University, Tian shui, Gansu,
P.R.China
Abstract Two variants of the Weyl spectrum are discussed We nd for example that if
one of them coincides with the Drazin spectrum then generalized Weyls theorem holds, and
conversely for isoloid operators.
Keywords Browders theorem, generalized Weyls theorem, operators.
. Introduction
H.Weyl examined the spectra of all compact perturbations of a hermitian operator on
Hilbert space and found in that their intersection consisted precisely of those points of the
spectrum which were not isolated eigenvalues of nite multiplicity. This Weyls theorem has
since been extended to hyponormal and to Toeplitz operators Coburn , to seminormal and
other operators Berberian , and to Banach spaces operators Istratescu , Oberai
. Variants have been discussed by Harte and Lee and Rakocevic , M.Berkani and
J.J.Koliha . In this note we show how generalized Weyls theorem follows from the equality
of the Drazin spectrum and a variant of the Weyls spectrum.
Recall that the Weyls spectrum of a bounded linear operator T on a Banach space X
is the intersection of the spectra of its compact perturbations
w
T
T K K KX .
Equivalently
w
T i T I fails to be Fredholm of index zero. The Browder spectrum
is the intersection of the spectra of its commuting compact perturbations
b
T
T K K KX commT .
Equivalently
b
T i T I fails to be Fredholm of nite ascent and descent. the Weyls
theorem holds for T i
T
w
T
T,
where we write
T iso T lt dimNT I lt
for the isolated points of the spectrum which are eigenvalues of nite multiplicity. Harte and
Lee have discussed a variant of Weyls theorem the Browders theorem holds for T i
T
w
T
T.
Vol. Browders theorem and generalized Weyls theorem
What is missing is the disjointness between the Weyl spectrum and the isolated eigenvalues
of
nite multiplicity equivalently
w
T
b
T.
For a bounded linear operator T and a nonnegative integer n dene T
n
to be the restriction
of T to RT
n
viewed as a map from RT
n
into RT
n
in particular T
T. If for
some integer n the range space RT
n
is closed and T
n
is upper resp.a lower semiFredholm
operator, then T is called an upper resp.lower semiBFredholm operator. Moreover if T
n
is
a FredholmWeyl or Browder operator, then T is called a BFredholm BWeyl or BBrowder
operator. Similarly, we can dene the upper semiBFredholm, BFredholm, BWeyl, and B
Browder spectrums
SF
T,
BF
T,
BW
T,
BB
T. A semiBFredholm operator is an
upper or a lower semiBFredholm operator.
See Let T BX and let
T n N m N, m n RT
n
NT RT
m
NT.
Then the degree of stable iteration disT of T is dened as disT inf T.
Let T be a semiBFredholm operator and let d be the degree of the stable iteration of T.
It follows from , Proposition . that if T
d
is a semiFredholm operator, and indT
m
indT
d
for each m d. This enables us to dene the index of a semiBFredholm operator T
as the index of the semiFredholm operator T
d
.
In the case of a normal operator T acting on a Hilbert space, Berkani , Theorem .
showed that
BW
T TET,
ET is the set of all eigenvalues of T which are isolated in the spectrum of T. This result gives
a generalization of the classical Weyls theorem. We say T obeys generalized Weyls theorem
if
BW
T TET , Denition ..
In this paper, we describe Browders theorem and generalized Weyls theorem using two
new spectrum sets which we dene in section .
. Browders theorem and generalized Weyls theorem
Using Corollary . in , we can say that
BB
T
D
T, where
D
TT
is not a pole of T . We call
D
T the Drazin spectrum of T. We can prove that the Drazin
spectrum satises the spectral mapping theorem, and the Drazin spectrum of a direct sum is
the union of the Drazin spectrum of the components.
In this section, our rst result is
Theorem .. Browders theorem holds for T if and only if
BW
T
D
T.
Proof. Suppose that Browders theorem holds for T. We only need to prove that
D
T
BW
T. If
is not in
BW
T, then T
I is BWeyl operator, and, in particular, an operator
of topological uniform descent. , Remark iii asserts that there exists gt such that T I
Junhong Tian and Wansheng He No.
is Weyl if lt
lt . Since Browders theorem holds for T, it follows that T I is
Browder operator if lt
lt . Then
is a boundary of T. , Corollary .. tells
us that
is not in
D
T. Conversely, Suppose that
BW
T
D
T. We need to prove
that
b
T
w
T. Suppose that T
I is Weyl. Then
is not in
D
T, which means
that
isoT. Thus T
I is Browder. This proves that Browders theorem holds for T.
Theorem .. If Browders theorem holds for T BX and S BX, and p is a
polynomial, then Browders theorem holds for
pT p
BW
T
BW
pT
Browders theorem holds for
TS
BW
TS
BW
T
BW
S.
Proof. Browders theorem holds for pT if and only if
BW
pT
D
pT
p
D
Tp
BW
T and Browders theorem holds for T S if and only if
BW
TS
D
TS
D
T
D
S
BW
T
BW
S.
Theorem .. If T BX, then
indT IindT I for each pair, C
e
T
if and only if
p
BW
T
BW
pT for each polynomial p.
Proof. By , Remark iii, indT IindT I for each pair, C
e
T if
and only if indT IindT I for each pair , C
BF
T. From , Corollary .
and , Theorem ., the spectral mapping theorem for the BWeyl spectrum may be rewritten
as the implication, for arbitrary n N and
i
C,
T
IT
IT
n
I B Weyl T
j
I B Weyl for each j , , , n.
Now if indT I on C
BF
T, then we have
n
j
indT
j
I ind
n
j
T
j
I indT
j
I j , , , n,
and similarly if indT I on C
BF
T. If conversely there exist , C
e
T for
which indT I m lt lt k indT I, then pT T I
k
TI
m
is a Weyl
operator whose factors are not BWeyl. It is a contradiction. The proof is completed.
We turn to a variant of the Weyl spectrum, involving a condition introduced by Saphar
and the zero jump condition of Kato . Let
T C there exists gt such that T I is Weyl and
NT I
n
RT I
n
if lt lt
and let
D
TC
D
T, T CT,
TC
T. Then T
D
T
T.
Vol. Browders theorem and generalized Weyls theorem
T is called isoloid if isoT NT I .
Theorem .. T BX is isoloid and generalized Weyls theorem holds for T if and
only if
T
D
T.
Proof. Suppose T is isoloid and generalized Weyls theorem holds for T. We only need to
prove
D
T
T. Let
D
T and suppose
T. Then there exists gt such
that T I is Weyl and NT I
n
RT I
n
if lt
lt . Since generalized
Weyls theorem implies Weyls theorem for T , Theorem ., it follows that T I is
Browder and therefor NT I NT I
n
RT I
n
for lt
lt ,
which means that T I is invertible if lt
lt . Then
is an isolated point in T.
Thus
ET T
BW
T because T is isoloid. , Corollary . asserts that
is not
in
D
T, it is a contradiction.
Conversely, Suppose
T
D
T. By ET
T
D
T, we get ET
T
BW
T. Conversely, let
T
BW
T, that is T
I is BWeyl, then there
exists gt such that T I is Weyl and NT I
n
RT I
n
if lt
lt ,
then
T
D
T. Thus
ET. In the following, we will prove T is isoloid. Let
isoT, then
T
D
T, thus
is a pole of T, so NT
I , which
means that T is isoloid.
Corollary .. Suppose T, S BX are all isoloid. If generalized Weyls theorem holds
for T and S and if p is a polynomial, then generalized Weyls theorem holds for
pT
pT p
T
and generalized Weyls theorem holds for
TS
TS
T
S.
Proof. If T and S are isoloid, then pT and T S are isoloid. Then generalized Weyls
theorem holds for pT
pT
D
pT p
D
Tp
T and generalized
Weyls theorem holds for T S
TS
D
TS
D
T
D
S
T
S.
In the following, we suppose that HT HT is the class of all complexvalued
functions which are analytic on a neighborhood region of T.
Theorem .. T BX, then
indT IindT I for each pair, C
e
T
if and only if
f
T
fT for anyf HT.
Proof. Suppose indT IindT I for each pair , C
e
T. For any f
HT, let
f
T and suppose
f
, where
T. If
is not in
fT,
then there exists gt such that fT I is Weyl and NfT I
n
RfT I
n
Junhong Tian and Wansheng He No.
if lt
lt . Then
k
fT, where
k
fT C RfT I is
closed and NfT I
n
RfT I
n
, which is dened by T.Kato in . In the
following we will prove that
T. By continuity of f, there exists gt such that
lt f
f
f
lt if lt
lt . Then fT f
I is Weyl and
NfT f
I
n
RfT f
I
n
. Thus f
is not in
k
fT f
k
T,
Satz . So
k
T, which means that NT
I
n
RT
I
n
. Clearly,
is not an
isolated point of T. Suppose
T satises lt
lt . Let h f f
.
Then h for all
k
T. Clearly, h has zeros in T. , Satz asserts now that h
has only a nite number of zeros in T. Let
,
,,
m
be these zeros
i
j
for i j
and n
,n
,n
,,n
m
be their respective orders. Then we can denote fT f
I by
fT f
IT
I
n
T
I
n
T
m
I
n
m
gT ,
where gT is invertible and
i
j
for i, j , , , , m. Since fT f
I is Weyl, it
follows that T
i
I is Fredholm and indfTf
I
m
i
indT
i
I
n
i
. Thus T
i
I
is Weyl. We now get that there exists gt such that T
I is Weyl and NT
I
n
RT
I
n
if lt
lt . Then
T. It is in contradiction to the fact
T. Then f
T
fT for any f HT.
For the converse, if there exist , C
e
T for which indT I m lt lt k
indT I, let fT T I
k
TI
m
. Then f
T but is not in
fT. It
is a contradiction. The proof is completed.
Corollary .. If T BX is isoloid and generalized Weyls theorem holds for T, then
the following statements are equivalent
indT IindT I for each pair , C
e
T
BW
fT f
BW
T for every f HT
generalized Weyls theorem holds for fT for every f HT
fT f
T for every f HT.
References
A.Aluthge and Derming Wang, whyponormal operators, Integr.equ.oper.theory,
,.
S.K.Berberian, An extension of Weyls theorem to a class of not necessarily normal
operators, Michigan Math. J., , .
S.K.Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. ,
.
M.Berkani and M.Sarih, On semi BFredholm operator, Glasgow Math.J., ,
.
M.Berkani, Index of BFredholm operators and generalization of a Weyls theorem,
Proc.Amer.Math.Soc., , .
Vol. Browders theorem and generalized Weyls theorem
M.Berkani and J.J.Koliha, Weyl type theorems for bounded linear operators,
Acta.Sci.Math.Szeged,
,.
M.Berkani, On a class of quasiFredholm operator, Integral Equations and operator
Theory, , .
L.A.Coburn, Weyls theorem for nonnormal operators, Michigan Math. J. ,
.
S.Grabiner, Uniform ascent and descent of bounded operators, J.Math.Soc. Japan,
,.
R.Harte and Woo Young Lee, Another note on Weyls theorem, Trans. Amer. Math.
Soc., , .
V.I.Istr atescu, On Weyls spectrum of an operator I, Rev. Ronmaine Math. Pure
Appl., , .
D.Kato, Perturbation theory for linear operator, SpringerVerlag New York Inc., .
T.Kato, Perturbation theory for nullity, deciency and other quantities of linear oper
ators, J.Anal.Math., , .
J.Ph.Labrousse, Les op erateurs quasiFredholm une g e n eralisation des op erateurs
semiFredholm, Rend.Circ.Math.Palermo, , No., .
D.C.Lay, Spectral analysis using ascent, descent, nullity and defect, Math.Ann., ,
.
M.Mbekhta and V.M uller, On the axiomatic theory of spectrum , Studia Math.,
, No., .
K.K.Oberai, On the Weyl spectrum, Illinois J. Math., , .
V.Rakocevic, Operators obeying aWeyls theorem, Rev. Roumaine Math. Pures
Appl., , .
P.Saphar, Contribution a letude des applications lineaires dans un espace de Banach,
Bull. Soc. Math. France, , .
C.Schmoeger, Ein spektralabbildungssatz, Arch.Math., , .
A.E.Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math.Ann.,
,.
H.Weyl,
Uber beschrankte quadratische Formen, deren Dierenz vollstetig ist, Rend.
Circ. Mat. Palermo, , .
Scientia Magna
Vol. , No. ,
On the instantaneous screw axes of two
parameter motions in Lorentzian space
Murat Kemal Karacan
and Levent Kula
Usak University, Faculty of Sciences and Arts,
Department of Mathematics, Usak, Turkey
Ahi Evran University, Faculty of Sciences and Arts,
Department of Mathematics, Kirsehir, Turkey
murat.karacanusak.edu.tr lkulaahievran.edu.tr
Abstract In this study two parameter motion is given by using the rank of rotation matrix
in Lorentzian space. It is shown locus of instantaneous screw axis is a ruled surface at any
position of , , .
Keywords Two parameter motion, one parameter motion, instantaneous screw axis
I.S.A, Lorentzian Space.
. Introduction and preliminaries
Let IR
n
r
,r
,,r
n
r
,r
, ..., r
n
IR be a ndimensional vector space, r
r
,r
,,r
n
and s s
,s
,,s
n
be two vectors in IR
n
, the Lorentz scalar product of r
and s is dened by
r, s
L
r
s
r
s
r
n
s
n
.
L
n
IR
n
,,
L
is called ndimensional Lorentz space, or Minkowski nspace. We denote
L
n
as IR
n
,,
L
. For any r r
,r
,r
,ss
,s
,s
L
, in the meaning Lorentz vector
product of r and s is dened by
r
L
sr
s
r
s
,r
s
r
s
,r
s
r
s
,
where e
L
e
e
,e
L
e
e
vector, a lightlike vector or a timelike vector if r, r
L
gt ,
r, r
L
or r, r
L
lt respectively. For r L
n
, the norm of r dened by r
L
r, r
L
,
and r is called a unit vector if r
L
. In the Minkowski nspace, the two parameter
motion of a rigid body is dened by
Y , A, X C, , .
where A SOn, is a positive semi orthogonal matrix, C IR
n
is a column matrix, Y and
X are position vectories of the same point B respectively, for the xed and moving space with
respect to semi orthonormal coordinate systems. The two parameter motion is given by .,
for , , , we have
A, A
,A
T
,I
Vol. On the instantaneous screw axes of two parameter motions in Lorentzian space
and
C, .
Then xed and moving space is coincided. If t, t, then one parameter motion is
obtained from two parameter motion. Since A SOn, , we have
A
T
, A, A, A
T
,I
n
, where
.
...
.....
...
..
nxn
.
For the sake of the short we shall take as A
T
,A
T
and A, A.
Denition . Taking the derivation with respect to t in equation Y , A, X
C, where let t and t, then it follows that
.
YY
.
Y
.
,
.
AA
.
A
.
,
.
CC
.
C
.
,
.
Y
.
AX
.
CA
.
X.
So
.
Y,
.
AX
.
C, A
.
X are called absolutely sliding and relative velocities of the point B has position
vectories
b , respectively. Let X be solution of system of
V
f
.
AX
.
C and the solution
is constant on the xed and moving space at position t. These points X is called instantaneous
pole points at every position t.
Denition . If rank
.
A n r be an even number on the two parameter motion
given by equation Y , A, XC, , then at any position of points the locus having
a velocity vector with stationary norm is a line. The line is called instantaneous screw axis
and
denoted by I.S.A. . Furthermore the moving space screw axis is dened by X P E
where P is a particular solution of equation
.
AX
.
C and E represent a bases of solution
space of homogeneous equation
.
AX .
. The instantaneous screw axes of two parameter motions
Theorem . Let A SOn, and let n be an odd number. Then the rank of A
and
A
are even.
Proof. Since
A
T
, A, A, A
T
,I
n
and A, A
T
, I, then
A
A
T
AA
T
,
I
n
Murat Kemal Karacan and Levent Kula No.
A
A
T
and
A
A
T
AA
T
A
A
T
.
Thus A
and A
are semi skewsymmetric matrices. Since n is an odd number it follows that
det A
,
det A
.
Thus it must be rank A
r even and rank A
r even.
Theorem . Let A SOn, . Then
rank A
rank A
and
rank A
rank A
.
Proof. Since
A, A
T
,I
n
.
it takes derivation with respect to , it follows that
A
A
T
AA
T
A
A
T
A
A
T
A
A
T
AA
T
A
A
T
A
A
T
AA
T
..
Since rank A
, we get A
and A
T
. We have following that .
A
A
T
A
A
T
.
For every x IR
n
, we have
A
A
T
x
T
x
T
A
A
T
x
T
x
A
A
T
x
T
xA
xA
T
.
So that xA
, xA
L
from the nondegenere property, xA
. Since it is true for every
x IR
n
, so we get A
and rank A
. It takes derivation with respect to in the
equation . similarly it follows that
A
A
T
AA
T
,
A
A
T
A
A
T
A
A
T
AA
T
,
A
A
T
A
A
T
AA
T
.
Vol. On the instantaneous screw axes of two parameter motions in Lorentzian space
Since rank A
, we get A
and A
T
, thus we have
A
A
T
A
A
T
.
Since it is true for every x IR
n
, we have the following
A
A
T
x
T
x
T
A
A
T
x
T
x
A
A
T
x
T
xA
xA
T
.
Hence
xA
, xA
L
xA
.
For every x IR
n
it is true, we get A
and rank A
. Conversely it is obviously to
see.
. Special case n
Since A
and A
are semi skewsymmetric matrices especially
A
j
j
j
j
j
j
,A
i
i
i
i
i
i
.
Let A
A
T
,A
A
T
. The equation X A
, Y , is obtained from the
equation of Y , A, X. By dierentiating the equation Y , A, X with
respect to t , we have
Y
.
Y
.
A
.
A
.
X
A
.
A
.
A
,Y,.
In the position , , , we have
Y
.
Y
.
A
.
A
.
Y,
Y,.
Since A
and A
are semi skewsymmetric matrices, we get
j
.
i
.
j
.
i
.
j
.
i
.
j
.
i
.
j
.
i
.
j
.
i
.
Murat Kemal Karacan and Levent Kula No.
and also angular velocity matrix is an semi skewsymmetric. Since A matrix is semi
orthogonal,
we have the following equation
A, A
T
, I.
Now dierentiating with respect to t, it follows that
A
.
A
.
A
T
, A,
A
T
.
A
T
.
.
For , , , we obtain
A
.
A
.
A
.
A
.
T
.
Since A
.
A
.
, it follows that
T
,
where is semi skewsymmetric matrix. Since pole points which are the points of sliding
velocity is zero given by
V
f
A
.
A
.
X
C
.
C
.
is pole points of two parameter motion by
Y , A, X C, .
The equation
A
.
A
.
X
C
.
C
.
.
can be solution it must be rank
A
.
A
.
rank . It follows that
Y , A, X C,
A, X Y , C,
A
, A, X A
, Y , C,
XA
, Y , C, .
If we get write this value of X in the equation ., we have
A
.
A
.
A
, Y , C,
C
.
C
.
..
In the position , , , we have
A
.
A
.
A
,
A
.
A
.
.
And if we say
Y , C, Y
,,
then the equation . form
L
Y
,
C
.
C
.
..
Vol. On the instantaneous screw axes of two parameter motions in Lorentzian space
The solution of equation . gives xed pole points of the motion. For the solution of equation
. it must be veried the condition
,C
.
C
.
L
.
In generally, this condition cant verify. But we can separate two composite velocities
C
.
C
.
either orthogonal
or parallel. If W is angular velocity matrix of moving space,
then it can nd W A
, . Since A
, I, we have W in the position
,,.
Let
E
L
and
E
W
W
L
.
Then we have E
AE, where IR. For , , and , it follows that
E
E.
We can separate two components of the velocity C
.
C
.
which form
U
C
.
C
.
,C
.
C
.
L
,
L
and
V
,C
.
C
.
L
,
L
,
one is orthogonal to vector
, another is parallel to vector
respectively, in which
C
.
C
.
C
.
C
.
,C
.
C
.
L
,
L
,C
.
C
.
L
,
L
.
That is
,U
L
,
C
.
C
.
,C
.
C
.
L
,
L
L
,C
.
C
.
L
,C
.
C
.
L
,
L
,
L
,C
.
C
.
L
,C
.
C
.
L
and
V
,C
.
C
.
L
,
L
.
Murat Kemal Karacan and Levent Kula No.
If we write velocity U replacing C
.
C
.
in equation ., we have this condition , U
L
is veried in this equation. Let rank ,then the system can be solution. Now we get the
solution of the equation
L
Y
,U.
Here a
L
b
L
c
a,
b
L
c
a,
c
L
b . It follows that
L
Y
,U,
L
L
Y
,U,
L
L
Y
,
L
U,
,
L
Y
,,Y
,
L
L
Y,
Y
,
,Y
,
L
,
L
L
U
,
L
,
Y
,
L
U
,
L
.
If we write the last equation replacing by the equation
Y
, Y , C, ,
so we have
Y,
L
U
,
L
C, .
Hence we get
Y , Q , IR. .
This means that it is a line which is passes though point Q and straight . The line is called
xed pole axis in the xed space, the expression of the xed pole axis in the moving space nd
to write instead of value Y , in the equation ., it follows that
L
U
,
L
C, A, X C, ,
L
U
,
L
A, X,
XA
,
L
U
,
L
A
,.
In the position , , , it follows that
X
L
U
,
L
,
XP.
If the pole axis of xed and moving space coincide in the position , , , we have P Q
and C, . Thus lines passes though points Q and P is straight
, which are the pole
Vol. On the instantaneous screw axes of two parameter motions in Lorentzian space
axis in xed and moving space. In the position , of motion the lines have a velocity vector
with stationary norm, locus of the lines which called instantaneous screw axis. The equations
Y Q , IR
X P , IR
depend only on parameters
.
and
.
. Thus there is
the one parameter motion. There are
instantaneous screw axis since the parameters
.
and
.
depend only on t . The locus of
this screw axis is a ruled surface. Indeed the following equations determine a ruled surface,
Y t, Qt t, IR
Xt, Pt t, IR.
References
Bottema O., Roth B., Theotical Kinematics, North Holland Publ. Com., .
Hacisalihoglu H.H., On The Geometry of Motion In The Euclidean nSpace, Commu
nications. Ank.,
Univ. Seri A., , .
Karacan M. K., Kinematic Applications of TwoParameter Motion, Ankara University,
Graduate School and Natural Sciences, Ph.D Thesis, .
Karacan M.K., Yayli Y., On The Instantaneous Screw Axes of Two Parameter Motions,
Facta Universitatis, Series, Mechanics, Automatic, Control and Robotics, , No., .
ONeill B., SemiRiemannian Geometry with Applications to Relativity, Academic
Press, London, .
Scientia Magna
Vol. , No. ,
Cyclic dualizing elements in Girard quantales
Bin Zhao
and Shunqin Wang
College of Mathematics and Information Science, Shaanxi Normal University,
Shaanxi, Xian , P.R. China
School of Mathematics and Statistics, Nanyang Normal University,
Henan, Nanyang , P.R. China
zhaobinsnnu.edu.cnmath.wangsq.com
Abstract In this paper, we study the interior structures of Girard quantale and the cyclic
dualizing elements of Girard quantale. some equivalent descriptions for Girard quantale are
given and an example which shows that the cyclic dualizing element is not unique is given.
Keywords Complete lattice, quantale, Girard quantale, cyclic dualizing element.
. Preliminaries
Quantales were introduced by C.J. Mulvey in with the purpose of studying the spectrum
of C
algebras and the foundations of quantum mechanics. The study of such partially ordered
algebraic structures goes back to a series of papers by Ward and Dilworth , in the s.
It has become a useful tool in studying noncommutative topology, linear logic and C
algebra
theory . Following Mulvey, various types and aspects of quantales have been considered
by many researchers . The importance of quantales for linear logic is revealed in Yetters
work . Yetter has claried the use of quantales in linear logic and he has introduced the
term Girard quantale. In , J. Paseka and D. Kruml have shown that any quantale can be
embedded into a unital quantale. In , K.I. Rosenthal has proved that every quantale can
be embedded into a Girard quantale. Thus, it is important to study Girard quantale. This is
the motivation for us to investigate Girard quantale. In the note, we shall study the interior
structures of Girard quantale and the cyclic dualizing element in Girard quantales.
We use to denote the top element and the bottom element in a complete lattice. For
notions and concepts, but not explained, please to refer to .
Denition .. A quantale is a complete lattice Q with an associative binary operation
amp satisfying
aamp
b
aampb
and
b
ampa
b
ampa
for all a Q, b
Q.
This work was supported by the National Natural Science Foundation of ChinaGrant No. and
the Research Award for Teachers in Nangyang Normal University, China nynu
Vol. Cyclic dualizing elements in Girard quantales
An element e Q is called a unit if aampe eampa a for all a Q. Q is called unital if Q
has the unit e.
Since aamp
and
ampa preserve arbitrary sups for all a Q, they have right adjoints
and we shall denote them by a
r
and a
l
respectively.
Proposition .. Let Q be a quantale, a, b, c Q. Then
aampa
r
bb
a
r
b
r
c bampa
r
c
Again, analogous results hold upon replacing
r
by
l
.
Denition .. Let Q be a quantale, An element c of Q is called cyclic, if a
r
ca
l
c for all a Q. d Q is called a dualizing element, if a a
l
d
r
da
r
d
l
d
for all a Q.
Denition .. A quantale Q is called a Girard quantale if it has a cyclic dualizing
element d.
Let Q be a Girard quantale with cyclic dualizing element d and a, b Q, dene the binary
operation by ab a
ampb
, then we can prove that a
and
a preserve arbitrary infs
for all a Q, hence they have left adjoints and we shall denote them by a
r
and a
l
respectively. If a
r
da
l
d, we shall denote it by a d, or more frequently by a
if
d is a cyclic dualizing element.
. The equivalent descriptions for Girard quantale
In this section, we shall study the interior structures of Girard quantale and give some
equivalent descriptions for Girard quantale. According to the above, we know that there are
six
binary operations on a Girard quantale such as amp ,
r
,
l
,,
r
,
l
,
we shall respectively call them multiplying, right implication, left implication, Par operation,
dual right implication and dual left implication for convenience.
Theorem .. Let Q be a unital quantale,
Q Q an unary operation on Q. Then
Q is a Girard quantale if and only if
a
l
b aampb
a
r
bb
ampa
.
Proof. The necessity is obvious. Suciency suppose and hold, a Q. Denote
the unit element by e on Q, then a e
l
a eampa
a
. Thus a a
, hence
a
l
e
aampe
aampe
a
.
Similarly we get a
a
r
e
. Take d e
, thus d is a cyclic element of Q. Again
a Q, a e
e
a
e
a
a. This proves d e
is a dualizing
element on Q. Thus the proof is completed.
Theorem .. Let Q be a complete lattice.
r
QQ Q is a binary operation
on Q, a
r
Q Q and
r
aQQ
op
preserve arbitrary sups for all a Q.
Q Q is a unary operation on Q, e Q. For all a, b, c Q,
e
r
aaa
r
e
a
a
aabb
a
a
r
b
r
ca
r
b
r
c
Bin Zhao and Shunqin Wang No.
ca
r
b
ab
r
c
.
Then Q is a Girard quantale and
r
is the right implication operation.
Proof. Dene the binary operation aampb b
r
a
for all a, b Q.
amp satises associative law In fact, for all a, b, c Q,
aampbampc b
r
a
ampc c
r
b
r
a
.
Using the condition , we have c
r
b
r
a
c
r
b
r
a
. By the
denition of the binary operation amp we can get
aampbampc aampc
r
b
c
r
b
r
a
.
So aampbampc aampbampc.
Using the condition , we have aampb c b
r
a
cc
b
r
a
ba
r
c for all a, b, c Q.
For any a Q, b
i
iI
Q. If I , then aamp a
r
a
r
, since
again a
r
a
r
aamp , the last inequality obviously holds. So
aamp . Thus aamp preserves emptysups. If I , then
aamp
iI
b
i
iI
b
i
r
a
iI
b
i
r
a
iI
b
i
r
a
iI
aampb
i
.
Hence aamp preserves arbitrary sups for all a Q. Similarly, we can prove ampa preserves
arbitrary
sups for all a Q. Thus Q, amp is a quantale.
In accordance with the condition , we know e is the unit element corresponding to amp
on Q and a
r
e
a
. Denote by a
l
the right adjoint of ampa. Then
a
l
e
x Qx a
l
e
x Qxampa e
x Qa
r
x
e
x Qe a
r
x
x Qaampe x
x Qa x
x Qx a
a
.
This show e
is a cyclic element in Q. Using conditions and we know e
is also a
dualizing element on Q. Hence Q, amp, is a Girard quantale. We can easily prove
r
is
the right implication operation on Q by the above consideration.
Theorem .. Let Q be a complete lattice.
r
QQ Q is a binary operation
in Q, a
r
Q Q and
r
aQ
op
Q preserve arbitrary sups for all a Q.
Q Q is an unary operation in Q, d Q. For all a, b, c Q,
Vol. Cyclic dualizing elements in Girard quantales
d
r
aaa
r
d
a
a
aabb
a
a
r
b
r
ca
r
b
r
c
ca
r
bb
c
r
a
.
Then Q is a Girard quantale and
r
is the dual right implication operation.
Proof. Dene binary operation aampb b
r
a for all a, b Q,
i The binary operation amp is associative Since a, b, c Q,
aampbampc b
r
aampc
c
r
b
r
a
c
r
b
r
a
aampc
r
b
aampbampc.
ii Using the condition , we can prove
iI
a
i
iI
a
i
iI
a
i
iI
a
i
for any set I and a
i
iI
Q.
iii For all a Q, b
i
iI
Q, we have
aamp
iI
b
i
iI
b
i
r
a
iI
b
i
r
a
iI
b
i
r
a
iI
aampb
i
.
Similarly we have
iI
b
i
ampa
iI
b
i
ampa. Hence Q, ampis a quantale. Since aampd
d
r
ad
r
aad
ampb b
r
d
b, thus Q, amp is a unit quantale
with unit element d
.
iv If a Q, we have
a
l
d
x Qx a
l
d
x Qxampa d
x Qa
r
xd
x Qd
r
ax
x Qx a
a
.
Similarly, a
r
da
, hence d is a cyclic element in Q. d is also a dual element in Q by
condition . Thus Q, amp is a Girard quantale with cyclic dual element d. We easily know
r
is the dual right implication operation in Q by the denition of amp.
Obviously, Theorem . and Theorem . also hold if
r
and
r
are substituted by
l
and
l
respectively, right and left replace each other.
Theorem .. Let Q be a unital quantale with a unary operation
satisfying the
condition
CN a
a and a
r
bb
l
a
for all a, b Q. Then Q is a Girard quantale.
Bin Zhao and Shunqin Wang No.
. The cyclic dualizing element of Girard quantale
According to the denition of Girard quantale, we know that the cyclic dualizing element
plays an important role in Girard quantale, so we shall discuss the cyclic dualizing element in
this section. We shall account for whether the cyclic dualizing element is unique in a Girard
quantale when it is unique whether these Girard quantales determined by dierent cyclic
dualizing elements are dierent. Let us see the following example
Example .. Let Q , a, b, c, , the partial order on Q be dened as Fig , the
operator amp on Q be dened by Table . Then we can prove that Q is a commutative Girard
quantale. And we can prove that a, b and c are cyclic dualizing elements of Q.
d
d
d
d
Fig
a
b
c
amp
a
b
c
a
b
ca
b
ca
b
c
a
b
c
Table
Proposition .. Let Q be a unital quantale with the unit element e.
and
satisfy
the condition CN in Theorem .. Then e
e
if and only if
.
Proposition .. Let Q be a quantale, d
,d
are cyclic dualizing elements of Q,
,
are unary operations on Q induced by d
,d
respectively. Then d
d
if and only if
.
Theorem .. Let Q be a Girard quantale. Then there is a onetoone correspondence
between the set of cyclic dualizing elements in Q and the set of unary operations satisfying
the
condition CN in Theorem ..
Proposition .. Let Q be a Girard quantale. If is a cyclic dualizing element of Q,
then Q is strictly twosided.
Proof. Assume is a cyclic dualizing element in Q. Then
is the unit
of Q, hence a Q, aamp ampa a, this nished the proof.
Proposition .. If Q is a twosided Girard quantale, then the unique cyclic dualizing
element is the least element .
Proof. If Q is a twosided Girard quantale, then we have a aampe aamp a for all
a Q. Similarly, we have ampa a. Thus Q is strictly twosided. Suppose d is a cyclic dual
element in Q,
is the unary operation induced by d, then we have d d
. the
proof is nished.
Corollary .. Let Q be a Girard quantale with cyclic dualizing element . Then the
cyclic dualizing element of Q is unique.
Theorem .. Any complete lattice implication algebra is a Girard quantale with unique
cyclic dualizing element .
According the above conclusions, we have a question Whether the cyclic dualizing element
must be the least element if a Girard quantale has an unique cyclic dualizing element. The
Vol. Cyclic dualizing elements in Girard quantales
answer is negative. Let us see the following example.
Example .. Let Q , e, , the partial order on Q be dened by lt e lt , the
binary operation amp be dened by Table
amp
e
e
e
Table
It is immediate to verify Q being a Girard quantale with the unique cyclic dualizing element
e.
Question .. What is the necessary condition when the cyclic dualizing element of
Girard quantale is unique
References
C. J. Mulvey, Suppl. Rend. Circ. Mat. Palermo Ser. II, , .
M. Ward, Structure residation, Ann. Math., , .
R. P. Dilworth, Noncommutative residuated lattices, Trans. Amer. Math. Soc.,
,.
B. Banaschewski and M. Ern e, On Krulls separation lemma, Order, , .
J. Y. Girard, Linear logic, Theoret. Comp. Sci., , .
F. Borceux and G. Van den Bossche, Quantales and their sheaves, Order, ,
.
D. Kruml, Spatial quantales, Appl. Categ. Structures, , .
S. Q. Wang, B. Zhao, Prequantale congruence and its properties, Advances in Mathe
maticsIn Chinese, , .
S. W. Han, B. Zhao, The quantale completion of ordered semigroup, Acta Mathematica
SinicaIn Chinese, , No., .
D. Yetter, Quantales and noncommutative linear logic, J. Symbolic Logic, ,
.
J. Paseka, D. Kruml, Embeddings of quantales into simple quantales, J. Pure Appl.
Algebra, , .
K. I. Rosenthal, Quantales and their applications, Longman Scientic and Technical,
London, .
Scientia Magna
Vol. , No. ,
Compact spaces
S. Balasubramanian
, P. Aruna Swathi Vyjayanthi
, C. Sandhya
Department of Mathematics, Government Arts College Autonomous, KarurT.N.India
Department of Mathematics, C.R. College, ChilakaluripetA.P. India
Department of Mathematics, C.S.R. Sharma College, Ongole A.P. India
maniredimail.com vyju
redimail.com sandhya
karavadiyahoo.co.uk.
Abstract In this paper compactness and Lindeloness in topological space are intro
duced, obtained some of its basic properties and interrelations are veried with other types of
compactness and Lindeloness.
Keywords compact, Lindelo spaces
. Introduction
After the introduction of semi open sets by Norman Levine various authors have turned
their attentions to this concept and it becomes the primary aim of many mathematicians to
examine and explore how far the basic concepts and theorems remain true if one replaces
open
set by semi open set. The concept of semi compactness was introduced by C. Dorsett in .
After him Reilly and Vamanamurthy studied about semi compactness during . U.N. B.
Dissanayake and K. P. R. Sastry introduced locally Lindelo spaces. In the present paper we
introduce the concepts of compactness and lindeloness using open sets in topological
spaces.
Throughout the paper a space X means a topological space X, . The class of open
sets is denoted by OX, respectively. The interior, closure, interior, closure are
dened by A
o
,A
,A
o
,A
.
In section we discuss the basic denitions and results used in this paper. In sections
and we discuss about the compact and Lindeloness in the topological space and obtain
their basic properties.
. Preliminaries
A subset A of a topological space X, is said to be regularly open if A A
o
, semi
openregularly semi open or open if there exists an openregularly open set O such that
OAO
and closed if its complement is open. The intersection of all closed
sets containing A is called closure of A, denoted by A
. The class of all closed sets are
denoted by CLX, . The union of all open sets contained in A is called the interior
Vol. Compact spaces
of A,denoted by A
o
. A function f X, Y, is said to be continuous if the inverse
image of any openclosedset in Y is a openclosedset in X. irresolute if the inverse
image of any openclosedset in Y is a openclosed set in X. openclosed
if the image of every openclosedset is openclosed. homeomorphism if f is
bijective, irresolute and open. Let x be a point of X, and V be a subset of X, then V
is said to be neighbourhood of x if there exists a open set U of X such that x U V .
x X is said to be limit point of U i for each open set V containing x V Ux .
The set of all limit points of U is called derived set of U and is denoted by D
U.
Note . Clearly every regularly open set is open and every open set is semiopen
but the reverse implications do not holds good. that is, ROX OX SOX.
Theorem .. If x is a limit point of any subset A of the topological space X, ,
then every neighbourhood of x contains innitely many distinct points.
Theorem .. i union and intersection of any two open sets is not open.
ii Intersection of a regular open set and a open set is open.
iii If B X such that A B A
then B is open i A is open.
iv If A and R are regularly open and S is open such that R S R
. Then AR
AS.
Theorem .. In a semi regular space, int OX, generates topology.
Theorem .. i Let A Y X and Y is regularly open subspace of X then A is
open in X i A is open in
/Y
.
ii Let Y X and A OY,
/Y
then A OX, i Y is open in X.
iii Let Y X and Ais a neighborhood of x in . Then A is a neighborhood of x in
Y i Y is open in X.
Theorem .. An almost continuous and almost open map is irresolute.
Example . Identity map is irresolute.
Remark . For any topological space we have the following interrelations.
i compact nearlycompact almost compact weakly compact.
iicompact semicompact where none of the implications is reversible.
. Compact spaces
Denition .. A space X is said to be
i compact space if every open cover of it has a nite sub cover.
ii Countably compact space if every countable open cover of it has a nite sub
cover.
iii compact if it is the countable union of compact spaces.
Theorem .. Let X, be a topological space and A X. Then A is compact
subset of X i the subspace A,
/A
is compact.
Theorem .. i closed subset of a countably compact space is countably
compact.
ii A irresolute image of a countably compact space is countably compact.
iii countable product of countably compact spaces is countably compact.
S. Balasubramanian P. Aruna Swathi vyjayanthi, and C. Sandhya No.
iv countable union of countably compact spaces is countably compact.
Remark . countably compactness is a weakly hereditary property.
Theorem .. For a T
topological space X the following statements are equivalent
i X is countably compact.
iiEvery countable family of closed subsets of X which has the nite intersection prop
erty has a nonempty intersection.
iiiEvery innite subset has an accumulation point.
iv Every sequence in X has a limit point.
v Every innite open cover has a proper sub cover
Theorem .. Every irresolute map from a compact space into a T
space is
closed.
Proof. Suppose f X Y is irresolute where X is compact and Y is T
. Let
C be any closed subset of X. Then C is compact and so f C is compact. But then
f C is closed in Y by Theorem .. Hence the image of any closed set in X is closed
set in Y . Thus f is closed.
Theorem .. An continuous bijection from a compact space onto a T
space
is a homeomorphism.
Proof. Let f X Y be a continuous bijection from a compact space onto a T
space. Let G be an open subset of X. Then XG is closed and hence f XG is closed
by Th .. Since f is bijective fX G Y fG. Therefore f G is open in Y implies
f is open. Hence f is bijective irresolute and open. Thus f is homeomorphism.
Denition .. A space X is said to be Locally compact space if every x X has a
neighborhood whose closure is compact.
Note . Every compact space is locally compact.
Theorem .. If f X, Y, is irresolute, open and X is locally compact,
then so is Y.
Proof. Let y Y . Then x X fx y. Since X is locally compact x has a
compact neighborhood V Then by irresolute, open of f, fV is a compact neigh
borhood of y. Hence Y is compact.
Corollary . If f X, Y, is irresolute, open and X is compact, then Y
is Locally compact.
Proof. Obvious from above two theorems.
Theorem .. Let X, be a topological space and A X. Then A is locally compact
subset of X i the subspace A,
/A
is locally compact.
Theorem .. i closed subset of a locally Compact space is locally Compact.
ii countable product of locally Compact spaces is locally Compact.
iii countable union of locally Compact spaces is locally Compact.
Theorem .. For a topological space X, the following are equivalent
i X is compact.
ii Every family of closed subsets of X, having empty intersection has a nite subclass
with empty intersection.
Vol. Compact spaces
iii Every family of closed subsets of X which has the nite intersection propertyf.i.p
has a nonempty intersection.
Theorem .. Every compact, Hausdor space is almost regular.
Theorem .. Every pair of disjoint compact subsets of a Hausdor space have
disjoint open neighbourhoods.
From the denitions and remark , we have the following
Remark . For any topological space
nearlycompact compact semicompact but the converse is not true in general.
Weakening covering condition from nite cover to countable cover, we have the following
as a easy consequence of the above section.
. Lindelo and locally Lindelo spaces
In this section we dene Lindeloness using open sets their properties and characteri
zations are veried
Denition .. A space X, is said to be
i Lindelo space if every open cover of it has a countable sub cover.
ii Lindelo if it is the countable union of Lindelo spaces.
Theorem .. Let X, be a topological space and A X. Then A is Lindelo
subset of X i the subspace A,
/A
is Lindelo.
Theorem .. i closed subset of a Lindelo space is Lindelo.
ii countable product of Lindelo spaces is Lindelo.
iii countable union of Lindelo spaces is Lindelo.
Denition .. A space X, is said to be locally Lindelo space if every x X has
a Lindelo neighborhood.
Note . Every Lindelo space is locally Lindelo.
Theorem .. If f X, Y, is irresolute, open and X is locally Lindelo,
then so is Y .
Proof. Let y Y . Then x X f x y. Since X is locally Lindelo x has
a Lindelo neighborhood V Then by irresolute, open of f, fV is a Lindelo
neighborhood of y. Hence Y is Lindelo.
Corollary . If f X, Y, is irresolute, open and X is Lindelo, then Y
is Locally Lindelo.
Theorem .. Let X, be a topological space and A X. Then A is locally Lindelo
subset of X i the subspace A,
/A
is locally Lindelo.
Theorem .. i closed subset of a locally Lindelo space is locally Lindelo
ii countable product of locally Lindelo spaces is locally Lindelo.
iii countable union of locally Lindelo spaces is locally Lindelo.
Theorem .. For a Topological space X, the following are equivalent
i X is Lindelo.
ii Every family of closed subsets of X, having empty intersection has a countable
subclass with empty intersection.
S. Balasubramanian P. Aruna Swathi vyjayanthi, and C. Sandhya No.
iii Every family of closed subsets of X which has the countable intersection condi
tionc.i.c has a nonempty intersection.
Remark . For any topological space we have the following interrelations.
i Lindelo nearlyLindelo almost Lindelo weakly Lindelo.
iiLindelo semiLindelo where none of the implications is reversible.
Remark . For any topological space
nearlyLindelo Lindelo semiLindelo but the converse is not true in general.
Remark .
compact Lindelo
locally compact locally Lindelo.
none is reversible
Conclusion.
In this paper we dened new compact and lindelo axioms using open sets and studied
their interrelations with other compact and lindelo axioms.
Acknowlegement.
The Authors are thankful for the referees for their critical comments and suggestions for
the development of the paper.
References
Charless Dorsett, Semi compact R
, and Product spaces, Bull. Malayasian Math. Soc.,
, No., .
Charless Dorsett, Semi compactness, semi separation axioms and Product spaces, Bull.
Malayasian Math. Soc., , No., .
Dissanayake U.N.B. and Sastry K. P. R., Locally Lindelo spaces, I. J. P. A. M.,
, No., .
Dissanayake U.N.B. and Sastry K. P. R., A Remark on Lindelo spaces, I. J. P. A. M.,
, No., .
Norman Levine, Semi open sets and semi continuity in Topological spaces, Amer. Math.
Monthly, , .
Reilly I and Vamanamurthy M., On semi compact Spaces,Bull. Malayasian Math. Soc.,
, No., .
Sharma V.K., open sets, Acta Ciencia Indica, XXXII, .
Balasubramanian S., Sandhya C., and Aruna Swathi Vyjayanthi.P., open sets and
its properties communicated.
Scientia Magna
Vol. , No. ,
and sets of generalized Topologies
P. Sivagami
and D. Sivaraj
Department of Mathematics, Kamaraj College, Thoothukudi , Tamil Nadu, India
Department of Computer Science, D.J. Academy for Managerial Excellence,
Coimbatore , Tamil Nadu, India
Email ttn
sivagamiyahoo.co.in
Abstract We dene
sets,
sets, g.
sets and g.
sets and characterize these
sets for the collection , , , , b, of generalized topologies. For each , we
also dene and characterize the separation axioms T
i
, i , , and R
i
,i,.
Keywords open, closed, semiopen, preopen, open, open and bopen sets,
generalized topology.
. Introduction
In , Professor
A. Cs asz ar nicely presented the open sets and all weak forms of open
sets in a topological space X in terms of monotonic functions dened on X, the collection
of all subsets of X. For each such function , he dened a collection of subsets of X, called
the collection of open sets. A is said to be open if A A. B is said to be closed
if its complement is open. With respect to this collection of subsets of X, for A X, the
interior of A, denoted by i
A, is dened as the largest open set contained in A and the
closure of A, denoted by c
A, is the smallest closed set containing A. It is established
that is a generalized topology . In , semiopen sets are dened and discussed. In ,
open sets, preopen sets and open sets are dened and discussed. bopen sets are
dened in . If is the family of open sets, is the family of all semiopen sets, is
the family of all preopen sets, b is the family of all bopen sets and is the family of all
open sets, then each collection is a generalized topology. Since every topological space is
a generalized topological space, we prove that some of the results established for topological
spaces are also true for the generalized topologies , , , , b, . In section , we
list all the required denitions and results. In section , we dene the
and
operators
for each and discuss its properties. Then, we dene
sets,
sets, g.
sets and
g.
sets and characterize these sets. In section , for each , we dene and characterize
the separation axioms T
i
, i , , and R
i
,i,.
P. Sivagami and D. Sivaraj No.
. Preliminaries
Let X be a nonempty set and X X A B whenever A B.
For , a subset A X is said to be open if A A. The complement of a open
set is said to be a closed set. A family X is said to be a generalized topology if
and is closed under arbitrary union. The family of all open sets, denoted by , is a
generalized topology . A X, is said to be semiopen if there is a open set G such
that G A c
G or equivalently, A c
i
A , Theorem .. A is said to be preopen
if A i
c
A. A X, is said to be open if A i
c
i
A. A is said to be open
if A c
i
c
A. A is said to be bopen if A i
c
Ac
i
A. In , , and
, it is established that each is a generalized topology and so c
and i
can be dened,
similar to the the denition of c
and i
. In this paper, for , the pair X, is called a
generalized topological space or simply a space. For each , a mapping
XX
is dened by
A XXA. Clearly,
. The following lemmas will be useful in
the sequel. Moreover, one can easily prove the following already established results of
Lemma
..
Lemma .. Let X be a nonempty set, X and . Then the following hold.
aAc
A for every subset A of X.
b If A B, then c
Ac
B.
c For every subset A of X, x c
A if and only if there exists a open set G such that
A G For , it is established in Lemma . of .
d A is closed if and only if A c
A.
ec
A is the intersection of all closed sets containing A.
Lemma .. Let X be a nonempty set and X. Then X is semiopen ,
Proposition ..
. sets and sets
In this section, for the space X, , , we dene sets, sets, g. sets and
g. sets. For , sets and sets are dened in . For A X, we dene
A
U X A U and U and
A U X U A and U is closed . The
following Theorem . gives the properties of the operator
. Example . below shows that
the two sets in .e are not equal.
Theorem .. Let A, B and C
be subsets of X, and . Then the
following hold.
a If A B, then
A
B.
bA
A.
c
A
A.
d
C
C
.
e
C
C
.
f If A , then
A A.
g
Axc
xA.
Vol. and sets of generalized Topologies
hy
x if and only if x c
y.
i
x
y if and only if c
xc
y.
Proof.
a. Suppose x
B. Then there exists G such that B G and x G. Since
A B, there exists G such that A G and x G and so x
A which proves a.
b. The proof follows from the denition of
.
c. By b, A
A and so by a,
A
A. Let x
A. then there
exists G such that A G and x G which implies that
A G and x G. Therefore,
x
A which implies that
A
A. This completes the proof.
d Clearly, by a,
C
C
. Conversely, suppose x
C
. Then x
C
for every . Therefore, for every , there
exists G
such that C
G
and x G
. Let G G
. Then x G and
C
G which implies that x
C
. This completes the proof.
e The proof follows from a.
f The proof follows from the denition of
.
g Let x
A. If c
x A , then X c
x is a open set such that
A Xc
x and x Xc
x. Therefore, x
A, a contradiction to the assumption
and so c
x A . Hence
Axc
x A . Conversely, suppose for
x X, c
x A . If x
A, then there exists a open set G such that A G and
x G. Therefore, x X G which implies that c
xc
X G X G X A and
so c
x A , a contradiction. Therefore, x c
xA
A. This completes
the proof.
h Suppose y
x. Then y G whenever G is a open set containing x. Suppose
xc
y, then there is a closed set F such that y F and x F. Since X F is a
open set containing x, y F and so c
yc
F F which implies that c
yx
. By g, y
x, a contradiction. Hence x c
y.
Conversely, suppose x c
y. If y
x, there exists a open set G containing x such
that y G. Now y X G implies that c
yc
X G X G X x and so
c
y x which implies that x c
y, a contradiction to the hypothesis. Therefore,
y
x. This completes the proof.
i Suppose
x
y. Assume that z
x and z
y. Then by h
and g, x c
z and y c
z and so c
xc
z and y c
z.
Therefore, c
x y which implies that c
xc
y.
Conversely, suppose c
xc
y. Assume that z c
x and z c
y. By h,
x
z and y
z and so
x
z and y
z which implies
that y
x . Therefore,
y
x which completes the proof.
Example .. Let X a, b and X X be dened by , a
b, b b, X X. Then , b, X. If A a, B b, then
A
X,
B B and
AB
. Since
A
B B,
A
B
A B.
The proof of the following Theorem . is similar to that of Theorem . and hence the
proof is omitted. Example . shows that the two sets in .d are not equal. Theorem .
P. Sivagami and D. Sivaraj No.
below gives the relation between the operators
and
.
Theorem .. Let A, B and C
be subsets of X, and . Then the
following hold.
a If A B, then
A
B.
b
A A.
c
A
A.
d
C
C
.
e
C
C
.
f If A is closed, then
A A.
Example .. Let X a, b, c, d and X X be dened by
, a b, b c, c b, d d, a, b b, c, a, c
b, a, d b, d, b, c b, c, b, d c, d, c, d b, d, a, b, c
b, c, b, c, d b, c, d, a, c, d X, a, b, d b, c, d and X X. Then
, d, b, c, a, c, d, b, c, d, X. If A a, B d, then
A a,
B
and
AB
a, d a, d. Since
A
B A,
A
B
AB.
If A a, c, d, then
A A but A is not closed. This shows that the reverse direction
of Theorem . f is not true.
If A c, d, then
A A but A is not open. This shows that the reverse direction
of Theorem .f is not true.
Theorem .. Let A be a subset of X, and . Then the following hold.
a
XAX
A.
b
XAX
A.
c
.
d
.
Proof. a and b follow from the denitions of
and
.
c If A X, then
AX
XA XX
A
A and so
.
d The proof is similar to the proof of c.
If X is a nonempty set, X and , a subset A of X is said to be a
set
if A
A and A is said to be a
set if A
A. In any space X, , the following
Theorem . lists out the
sets and the
sets.
Theorem .. Let X be a nonempty set, X and . Then the following hold.
a is a
set.
b A is a
set if and only if X A is a
set.
c X is a
set.
d The union of
sets is again a
set.
e The union of
sets is again a
set.
f The intersection of
sets is again a
set.
g The intersection of
sets is again a
set.
h If
, , , then X is a
set and so is a
set.
Proof.
a follows from Theorem .f since for every .
Vol. and sets of generalized Topologies
b Suppose A is a
set. Then A
A. Now X A X
A
X A,
by Theorem .b. Therefore, X A is a
set. The proof of the converse is similar with
follows from Theorem .a.
c follows from a and b.
d Let A
be a family of
sets. Therefore, A
A
for every .
Now
A
A
, by Theorem .d and so
A
A
.
e Let A
be a family of
sets. Therefore, A
A
for every .
Now A
A
A
by Theorem .d and so
A
A
by Theorem .b.
f Let A
be a family of
sets. Therefore, A
A
for every .
Now A
A
A
by Theorem .e and so
A
A
by Theorem .b.
g Let A
be a family of
sets. Therefore, A
A
for every .
Now
A
A
, by Theorem .e and so
A
A
.
h Since X by Lemma ., X for every
and so the proof follows from a
and b.
Remark .. Let
AXA
A and
AXA
A. Then
and
are topologies by Theorem ., such that arbitrary intersection of
open sets is a
open set and an arbitrary intersection of
open sets is a
open set.
Let X be a nonempty set, X and . A subset A of X is called a generalized
set in short, g.
set if
A F whenever A F and F is closed. B is called a
generalized
set in short, g.
set if X B is a g.
set. We will denote the family
of all g.
sets by D
and the family of all g.
sets by D
. The following Theorem .
shows that D
is closed under arbitrary union and D
is closed under arbitrary intersection.
Theorem . below gives a characterization of g.
sets.
Theorem .. Let X be a nonempty set, X and . Then the following hold.
a If B
D
for every , then B
D
.
b If B
D
for every , then B
D
.
Proof. a Let B
D
for every . Then each B
is a g.
set. Suppose F is
closed and B
F. Then for every , B
F and F is closed. By
hypothesis, for every ,
B
F and so
B
F. By Theorem .d,
B
F and so B
D
.
b Let B
D
for every . Then each B
is a g.
set and so X B
D
for every . Now X B
XB
D
, by a. Therefore,
B
D
.
Theorem .. Let X be a nonempty set, X and . Then a subset A of X is
a g.
set if and only if U
A whenever U A and U is open.
Proof. Suppose A is a g.
set. Let U be a open set such that U A. Then
XU is a closed set such that XU XA and so
XU
XA. Therefore,
XU
XA X
A and so U
A. Conversely, suppose the condition holds.
P. Sivagami and D. Sivaraj No.
Let A be a subset of X. Let F be a closed subset of X such that XA F. Then XF A
and so by hypothesis, X F
A. Then X
A F and so
X A F which
implies that X A is a g.
set. Therefore, A is a g.
set.
The remaining theorems in this section give some properties of g.
sets and g.
sets.
Theorem .. Let x X, X and . Then the following hold.
a x is either a open set or X x is a g.
set.
b x is either a open set or a g.
set.
Proof. a Suppose x is not a open set. Then X is the only closed set containing
X x and so
X x X. Therefore, X x is a g.
set.
b Suppose x is not a open set. By a, X x is a g.
set and so x is a
g.
set.
Theorem .. Let X be a nonempty set, X and . If B is a g.
set and
F is a closed set such that
B X B F, then F X.
Proof. Since B is a g.
set, X B is a g.
set such that X B F. Therefore,
XB F which implies that XF
B. Also,
B F and so XF X
B.
Hence X F
BX
B . Therefore, F X.
Corollary .. Let X be a nonempty set, X and . If B is a g.
set
such that
B X B is closed, then B is a
set.
Proof. By Theorem .,
B X B X and so X
BXB
which implies that X
B B . Therefore, B
B and so by Theorem .b,
B
B.
Theorem .. Let X be a nonempty set, X and . If A and B are subsets
of X such that A B
A and A is a g.
set, then B is a g.
set. In particular, if
A is a g.
set, then
A is a g.
set.
Proof. Since A B
A,
A
B
A
A and so
A
B. If F is any closed set such that B F, then A F and so
B
A F.
Therefore, B is a g.
set.
Corollary .. Let X be a nonempty set, X and . If A and B are subsets
of X such that
A B A and A is a g.
set, then B is a g.
set. In particular, if
A is a g.
set, then
A is a g.
set.
. Some separation axioms in generalized topological spaces
Let X be a nonempty set, X and . Then X, is called a T
space if
for all x, y X, x y, there exists a open set G such that x G and y G and there
exists a open set H such that x H and y H. T
space is dened in . The following
Theorem . gives a characterization of T
spaces in terms of closed sets and Theorem
. gives a characterization of T
spaces in terms of
sets.
Theorem .. Let X be a nonempty set, X and . Then X, is a T
space if and only if every singleton set is a closed set.
Proof. Suppose X, is a T
space. Let x X. If y X x, then x y. By
hypothesis, there exists a open set H
y
such that y H
y
and x H
y
and so y H
y
Xx.
Vol. and sets of generalized Topologies
Hence Xx H
y
y Xx is open and so x is closed. Conversely, suppose
each singleton set is a closed set. Let x, y X such that x y. Then X x and X y
are open sets such that y X x and x X y. Therefore, X, is a T
space.
Theorem .. Let X be a nonempty set, X and . Then X, is a T
space if and only if every subset of X is a
set.
Proof. Suppose X, is a T
space. Let A be subset of X. By Theorem .b,
A
A. Suppose x A. Then X x is a open set such that A X x and so
A X x. Hence every subset of X is a
set. Conversely, suppose every subset of
X is a
set and so
x x for every x X. Let x, y X such that x y. Then
y
x and x
y. Since y
x, there is a open set U such that x U
and y U. Similarly, since x
y, there is a open set V such that y V and x V.
Therefore, X, is a T
space.
Corollary .. Let X be a nonempty set, X and . Then the following are
equivalent.
a X, is a T
space.
b Every subset of X is a
set.
c Every subset of X is a
set.
Proof. a and b are equivalent by Theorem ..
b and c are equivalent by Theorem .b.
Theorem .. Let X be a nonempty set, X, and A X. Then the following
hold.
a If A is a
set, then A is a g.
set.
b If A is a
set, then A is a g.
set. The reverse directions are true if X, is a
T
space.
Proof.
a Suppose A is a
set. Then A
A. If A F where F is closed, then
A
A F. Therefore, A is a g.
set.
b The proof is similar to the proof of a.
The reverse directions follow from Corollary ..
Let X be a nonempty set, X and . X, is said to be a T
space if for
distinct points x and y in X, there exists a open set G containing one but not the other.
Clearly, every T
space is a T
space. Since every topology is a generalized topology
and in a topological space, T
spaces need not be T
spaces, T
spaces need not be T
spaces. Theorem . gives a characterization of T
spaces. The easy proof of Corollary .
is omitted.
Theorem .. Let X, be a space where X and . Then X, is a
T
space if and only if distinct points of X have distinct closures.
Proof. Suppose X, is a T
space. Let x and y be points of X such that x y.
Then there exists a open set G containing one but not the other, say x G and y G. Then
yc
x and so c
xc
y. Conversely, suppose distinct points of X have distinct
closures. Let x and y be points of X such that x y. Then c
xc
y. Suppose
zc
x and z c
y. If x c
y, then c
xc
y and so z c
y, a
P. Sivagami and D. Sivaraj No.
contradiction. Therefore, x c
y which implies that x X c
y and X c
y is
open. Hence X, is a T
space.
Corollary .. Let X, be a space where X and . Then X, is a
T
space if and only if for distinct points x and y of X, either x c
y or y c
x.
Let X be a nonempty set, X and . X, is said to be a T
space if for
distinct points x and y in X, there exist disjoint open sets G and H such that x G and
y H. Clearly, every T
space is a T
space and the converse is not true. Theorem .
below gives characterizations of T
spaces.
Theorem .. In a space X, , where X and , the following statements
are equivalent.
a X, is a T
space.
b For each x X and y x, there exists a open set U such that x U and y c
U.
c For every x X, x c
U x U and U is open.
Proof. ab. Let x, y X such that y x. Then, there exists disjoint open sets
U and H such that x U and y H. Then X H is a closed set such that U X H
and so c
U X H. U is the required open set such that x U and y c
U.
bc. Let x X. If y X such that x y, by b, there exists a open set U such that
x U and y c
U. Clearly, x c
U x U and U is open.
ca. Let x, y X such that y x. Then y x c
U x U and U is open,
by c. Therefore, y c
U for some open set containing x. U and X c
U are the
required disjoint open sets containing x and y respectively.
Let X be a nonempty set, X and . Then X, is said to be a R
space
if every open subset of X contains the closure of its singletons. X, is said to be a
R
space if for x, y X with c
xc
y, there exist disjoint open sets G and H
such that c
x G and c
y H. Clearly, every R
space is a R
space but the
converse is not true. The following Theorem . follows from Theorem .. Theorem . below
gives a characterization of R
spaces.
Theorem .. Let X be a nonempty set, X and . Then every T
space
is a R
space.
Theorem .. Let X be a nonempty set, X and . Then X, is a R
space if and only if every open subset of X is the union of closed sets.
Proof. Suppose X, is a R
space. If A is open, then for each x A, c
xA
and so cl
x x A A. It follows that A cl
x x A. Conversely, suppose A is
open and x A. Then by hypothesis, A B
where each B
is closed. Now
x A, implies that x B
for some . Therefore, c
xB
A and so X, is a
R
space.
Theorem .. Let X be a nonempty set, X and . Then X, is a R
space if and only if for x, y X, c
xc
y implies that c
xc
y.
Proof. Suppose X, is a R
space. Let x, y X, such that c
xc
y.
Suppose z c
x and z c
y. Since z c
y, there exists a open set G containing
z such that y G. Since z c
x, x G. Since y G, it follows that x c
y and so
x Xc
y. By hypothesis,c
x Xc
y and so c
xc
y . Conversely,
Vol. and sets of generalized Topologies
suppose the condition holds. Let G be a open set such that x G. If y G, then x y and
so x c
y which implies that c
xc
y. By hypothesis, c
xc
y and
so y c
x. Hence c
x G which implies that X, is a R
space.
Theorem .. Let X be a nonempty set, X and . Then X, is a R
space if and only if for x, y X,
x
y implies that
x
y.
Proof. Suppose X, is a R
space. Let x, y X such that
x
y.
Suppose that z
x
y. Then z
x and z
y. By Theorem .h,
xc
z and y c
z and so c
xc
z and c
yc
z . By Theorem
., c
xc
z and c
yc
z and so c
xc
y. By Theorem
.i,
x
y, a contradiction. Therefore,
x
y . Conversely,
suppose the condition holds. Let x, y X such that c
xc
y. Suppose that z
c
xc
y. Then z c
x and z c
y. By Theorem .h, x
z
and y
z and so
x
z and
y
z . By hypothesis,
x
z and
y
z and so
x
y. By Theorem .i,
c
xc
y, a contradiction. Therefore, c
xc
y . By Theorem ., X,
is a R
space.
Theorem .. Let X be a nonempty set, X and . Then the following are
equal.
a X, is a R
space.
b For any nonempty set A and a open set G such that A G , there exists a
closed set F such that A F and F G.
c If G is open, then G F F G and F is closed.
d If F is closed, then F G F G and G is open.
e For every x X, c
x
x.
Proof. ab. Suppose X, is a R
space. Let A be a nonempty set and G be a
open set such that AG . If x AG, then x G and so by hypothesis, c
x G.
If F c
x, then F is the required closed set such that A F and F G.
bc. Let G be open. Clearly, G F F G and F is closed. If x G, then
x G and so by b, there is a closed set F such that x F and F G which
implies that x F F G and F is closed. Therefore, G F F G and F is
closed. This completes the proof.
cd. Let F be closed. By c, X F K K X F and K is closed and so
F X K F X K and X K is open.
de. Let x X. If y
x, then by Theorem .g, x c
y . By d,
c
yGc
y G and G is open. Therefore, there is a open G such that
c
y G and x G which implies that y c
x. Therefore, c
x
x.
ea. Let G be a open set such that x G. If y c
x, then by e, y
x.
Since
x
G G, y G. Hence X, is a R
space.
Corollary .. Let X be a nonempty set, X and . Then the following are
equivalent.
a X, is a R
space.
b For every x X, c
x
x.
P. Sivagami and D. Sivaraj No.
Proof. ab. Let x X. By Theorem ., c
x
x. To prove the direction,
assume that y
x. By Theorem .h, x c
y and so c
xc
y which
implies that c
xc
y . By Theorem ., c
xc
y and so y c
x.
Hence c
x
x.
ba. The proof follows from Theorem ..
Theorem .. Let X be a nonempty set, X and . Then the following are
equivalent.
a X, is a R
space.
b For all x, y X, x c
y if and only if y c
x.
Proof. ab. Let x, y X such that x c
y. By Corollary ., x
y
and so by Theorem .h, y c
x. Thus x c
y implies that y c
x. Similarly,
we can prove that y c
x implies that x c
y.
ba. Conversely, suppose the condition holds. Let x X. If y c
x, then by hypothe
sis, x c
y and so by Theorem .h, y
x which implies that c
x
x.
By Theorem .e, X, is a R
space.
The following Theorem . gives a characterization of R
space and Theorem .
gives a characterization of T
space.
Theorem .. Let X be a nonempty set, X and . Then the following are
equivalent.
a X, is a R
space.
b For x, y X such that
x
y, there exist disjoint open sets G and H
such that c
x G and c
y H.
Proof. ab. Let x, y X such that
x
y. Then by Corollary .,
since X, is a R
space, c
xc
y and so there exists disjoint open sets G
and H such that c
x G and c
y H.
ba. Let x, y X such that c
xc
y. By Theorem .i,
x
y. By
hypothesis, there exist disjoint open sets G and H such that c
x G and c
yH
and so X, is a R
space.
Theorem .. Let X be a nonempty set, X and . Then the following are
equivalent.
aX, is a T
space.
bX, is both a R
space and a T
space.
cX, is both a R
space and a T
space.
Proof. ab. Suppose X, is a T
space. Clearly, X, is a T
space and
so singletons are closed sets, by Theorem .. If x, y X with c
xc
y, then
x y and so there exist disjoint open sets G and H such that x G and y H. Therefore,
c
x G and c
y H which implies that X, is a R
space.
bc. The proof is clear.
ca. Let x, y X such that x y. Since X, is a T
space, by Theorem .,
c
xc
y. Since X, is a R
space, there exist disjoint open sets G and H
such that c
x G and c
y H. Therefore, X, is a T
space.
Vol. and sets of generalized Topologies
References
A. Cs asz ar, Generalized Open Sets, Acta Math. Hungar., , No., .
A. Cs asz ar, On the interior and closure of a set, Acta Math. Hungar., ,
.
A. Cs asz ar, Generalized topology, generalized continuity, Acta Math. Hungar., ,
.
A. Cs asz ar, Generalized open sets in generalized topologies, Acta Math. Hungar.,
,.
A. G uld urdek and O.B.
Ozbakir, On semiopen sets, Acta Math. Hungar., ,
No., .
P. Sivagami, Remarks on interior, Acta Math. Hungar., , .
P. Sivagami and D. Sivaraj, Properties of some generalized topologies, Intern. J. General
Topology., , No., .
P. Sivagami and D. Sivaraj, On semiopen sets, Tamkang J. Math., ,
.
Scientia Magna
Vol. , No. ,
Generalized galilean transformations
and dual Quaternions
Yusuf Yayli
and Esra Esin Tutuncu
Ankara University, Faculty of Science,Department of Mathematics,
,TandoganAnkaraTURKEY
Rize University, Faculity of Education, RizeTURKEY
yayliscience.ankara.edu.tr tutuncu.eegmail.com
Abstract Quaternion operators have an important role as a screw and a rotation operator in
Euclidean motion. We introduce a new operator similar to quaternion operator in Galilean
motion. This operator is dened as a dual quaternion operator by using some information in
. Then, we have geneneralized this operator for ndimensional Galilean space.
Keywords Galilean transformation, dual complex numbers, Quaternion operators, lie group.
. Introduction
A unit real quaternion is a rotation operator on rigid body motion in Euclidean space. Unit
dual quaternions are also used both as rotation and screw operators. Unit dual quaternions
are
seen as screw operator especially in Mechanics and Kinematics. Galilean geometry of
motions
was studied in . n dimensional dual complex numbers was given in . These numbers are
viewed as analysis. Galilean transformations are given as shear motion on plane . Shear
motion in Galiean space G
was given . Moreover, union of shear motion and Euclidean
motions was introduced. Galilean transformation shear motion was given by quaternions
in dual quaternion form . Here, we redene dual quaternions in a new way for the rst
time. We work out Majerniks work in a new point of view by using structures of Lie groups
and algebras. These are subgroups of Heisenberg Lie groups. We obtain elements of groups
by
the exponential expansion of quaternion forms of an element of Lie algebra. And we
extended
the work to the Galilean space G
n
. Finally, we give Galilean transformation as dual quaternion
operators.
. Galilean transformations in galilean space G
Galilean transformations were examined widely in . Let X R
n
and G
n
be Galilean
space R
n
, with
X
x
,x
x
x
... x
n
,x
Vol. Generalized galilean transformations and dual Quaternions
for n , , . We redene Galilean transformation by using quaternion operators. For any
X x, y G
, Galilean transformation shear motion is dened as the following
fG
G
, X fX x, x y,
f is a linear transformation, so f has the matrices form as the following
fX
x
y
.
Lemma .. Let f be a linear transformation. Then f is a Galilean transformation, where
fG
G
, X fX x, x y.
Proof. For x and x , we have
X x fX
and
X y fX .
f is a Galilean transformation, because the linear function f is a isometry.
Theorem .. Let Gal be a Lie group. Then Gal and g are Lie algebras of
Gal, where
Gal
R
,g
R
.
. Dual numbers and galilean transformation in G
Every vector X x, y in R
can be written as X xy with
. So sp , R
.
The form of x y is called dual quaternion form of X.
Lemma .. Dual quaternion Q is a Galilean transformation in G
.
Proof. Since Q , we have
QX x y x y x
and
QX X .
So, Q is a Galilean transformation in G
. By using exponential map from Lie algebra to Lie
group, on g g as in g
form
e g Gal
ge
g
e
.
Corollary .. Q e
g
is a dual quaternion operator. Thus dual quaternion
operator Q is a Galilean transformation.
Yusuf Yayli and Esra Esin Tutuncu No.
. Galilean transformationshear motion in galilean space
G
In this section shear motion on Galilean spaces G
, Lie group structure of this motion and
exponential form are given.
Theorem .. f is a Galilean transformationshear motion, where
fG
G
X fX x, ax y, bx z
a
b
x
y
z
.
Proof. For x and x , we have
fX x X
and
fX
y
z
X.
So, f is a Galilean transformation.
Theorem .. Gal is a Lie group and the g is a Lie algebra of Gal, where
Gal
A
a
b
a, b R
,
g
a
a
b
a
,b
R
.
. Heisenberg lie group
The set of matrices
H
x
x
x
x
i
R, i , ,
is a Lie group with repect to the matrix multiplication. This Heisenberg group has many
important applications on SubRiemannian geometry and has very important role in physics.
Lemma .. Gal is a subgroup of Heisenberg Lie group.
Proof. Gal is obtained from Heissenberg Lie group by taking x
.
Vol. Generalized galilean transformations and dual Quaternions
. Dual Quaternions and galilean transformation in G
Every vector X in R
can be written as X x iy jz , where i
j
ij ji
and sp , i, j R
. The form X x iy jz is called as dual quaternion form of X R
.
Lemma .. Q ai bj is a Galilean transformation in G
.
Proof. Since Q ai bj, we have
QX ai bjx iy jz
x ax yi bx zj
and
QX X .
Thus the Q is a Galilean transformation.Furthermore, we can write g g as g a
ib
j.
So we can write an exponential map as follows
e g Gal
ge
g
e
a
ib
j
a
ib
j.
Corollary .. e
g
Qa
ib
j is a dual quaternion operator. Thus the dual
quaternion operator Q is a Galilean transformation.
. Galilean transformations in galilean space G
In this part shear motion is dened in Galilean spaces G
. Structure of Lie group of this
motion and exponential form are given.
Theorem .. f is a Galilean transformationshear motion, where
fG
G
X fX x, ax y, bx z, cx l
a
b
c
x
y
z
l
.
Proof. For x and x , we have
fX x X
and
fX
y
z
l
X.
So, f is isometry and Galilean transformation.
Yusuf Yayli and Esra Esin Tutuncu No.
Theorem .. Gal is a Lie group and g is a Lie algebra of the Gal, where
Gal
A
a
b
c
a, b, c R
,
g
a
a
b
c
a
,b
,c
R
.
. Dual Quaternions and galilean transformations in G
In this part, we reviewed study in . Every vector X in R
can be written as the form
X x iy jz kl , where
i
j
k
ij ji ik ki kj jk
and
sp , i, j, k R
.
The form X x iy jz kl is called as dual quaternion form of X R
.
Lemma .. Q ai bj dk is a Galilean transformation in G
.
Proof. Since Q ai bj dk , we have
QX ai bj dkx iy jz kl
x ax yi bx zj dx lk
and
QX X .
Thus, Q is a Galilean transformation. Furthermore, we can write g g as g a
ib
j
d
k. So we can write an exponential map as follows
e g Gal
ge
g
e
a
ib
jd
k
a
ib
jd
k.
Corollary .. Q e
g
a
ib
jd
k is a dual quaternion operator. Thus dual
quaternion operator Q is a Galilean transformation.
Vol. Generalized galilean transformations and dual Quaternions
. Galilean transformations in galilean space G
n
In this part, Galilean transformation generalized to Galilean spaces G
n
, by using Galilean
transformation in spaces G
,G
and G
.
Theorem .. f is a Galilean transformation, where
fG
n
G
n
X fX x
,
x
x
, ...,
n
x
x
n
.....
n
x
x
x
.
x
n
.
Proof. For x
and x
, we have
fX x
X
and
fX
x
x
x
n
X.
Then, f is isometry, thus f is a Galilean transformation.
Theorem .. Galn is a Lie group and gn is a Lie algebra of Gal, where
Galn
......
......
n
,
, ...,
n
R
,
gn
a
a
......
......
a
n
a
,a
,,a
n
R
.
Yusuf Yayli and Esra Esin Tutuncu No.
. Dual Quaternions and galilean transformation in G
n
In this part, every vector X in R
n
is written in form X x
x
i
x
i
x
n
i
n
,
where x
,x
,,x
n
are real numbers. The components of X and i
,i
,,i
n
are units
which satisfy the relations i
i
i
n
and i
j
i
k
i
k
i
j
, k, j n .
Also, sp , i
,i
,,i
n
R
n
,Xx
x
i
x
i
x
n
i
n
is called as dual
quaternion form of X R
n
.
Lemma .. Dual quaternion operator Q
i
i
n
i
n
is a Galilean
transformation.
Proof. Since Q
i
i
n
i
n
, we have
QX
i
i
n
i
n
x
x
i
x
i
x
n
i
n
x
x
x
i
x
x
i
n
x
x
n
i
n
and
QX X .
So, the dual quaternion operator Q is a Galilean transformation. For any element from Lie
algebra , a gn, a a
i
a
i
a
n
i
n
gn by using exponential map, we have
e gn Galn
ae
a
e
a
i
a
i
a
n
i
n
a
i
a
i
a
n
i
n
.
Corollary .. The transformation Q is a dual quaternion operator. So, dual quaternion
operator Q is a Galilean transformation.
Corollary .. Let a, b gn, then Qa e
a
Galn and Qb e
b
Galn.
Furthermore QaQb e
a
e
b
e
ab
Qa b. This implies the addition theorem for
the velocity on a Galilean transformation.
References
V. Majernik, Quaternion formulation of the Galilean spacetime Transformation, Acta
Phys, , No., .
I. M. Yaglom, A simple NonEuclidean geometry and its physical basic, SpringerVerlag
Inc, New York, .
M. Nadjakkah, A.R., Forough, galilean geometry of motions, arXiv.v, math.
DG, .
P. Fjelstad, S. G. Gal, NDimensional dual complex numbers advances in applied Cliord
algebras, , No., .
Scientia Magna
Vol. , No. ,
On fuzzy number valued Lebesgue outer
measure
T. Veluchamy
and P. S. Sivakkumar
sivaommurugaredimail.com
Dr. S. N. S Rajalakshmi College of Arts and Science Coimbatore, Tamil Nadu India
Department of Math. Government Arts College Coimbatore, Tamil Nadu India
Abstract This paper introduces the concept of fuzzy number valued Lebesgue outer
measure.
A nonnegative countably subadditive function m
on the power set of a set X by means of
a given additive function on an algebra of subsets of X, and a new collection of measurable
sets E are constructed , where E satisfy the relation m
Am
AE m
AXE ,
for any subset A of X .
Keywords Fuzzy number, fuzzy outer measure.
. Introduction
There are articles in the literature associated with fuzzy outer measure . We construct
fuzzy number valued outer measures and measurable sets that is similar to that of
Carathrodory
construction. In section we deal with fuzzy number valued Lebesgue outer measure in the
real
line R and in section , the results obtained in section are carried to arbitrary fuzzy number
valued measure space X, , m.
In section we give preliminary ideas relevant to fuzzy numbers.
. Basic Denitions
Denition .. Let F n n R , . For every n F, n is called a fuzzy number
if it satises the following properties
a n is normal i.e there exists an x R such that nx
b whenever , , the cut , n
x n is a closed interval denoted by
n
,n
We denote the set of all fuzzy numbers on R by F
.
Remark .. By decomposition theorem of fuzzy sets n
,
n
,n
.
Remark .. The fuzzy number a, b is dened by a, bx , i x a, b and a, bx
, i x / a, b. Similarly we can dene a, b.
If a b then a, b is simply denoted by a. i.e a is dened as ax , i x a and
ax i x a. Obviously then a, b, a F
.
T. Veluchamy and P. S. Sivakkumar No.
Further a, b
,
a, b. and a
,
a, a.
Remark .. A fuzzy number n which is increasing in the interval a, b, nb, c
and is decreasing in the interval b, c is simply denoted by a, b, c, d. Here nb, c means
nx for every x b, c.
The fuzzy number a, b, c, c where gt is denoted by a, b, c and the fuzzy number
a , a, b, c where gt is denoted by a, b, c.
Similarly the fuzzy number a, b, b, b where gt is denoted by a,
b and the fuzzy
number a , a, a, b where gt is denoted by a, b.
Denition .. For every a,
bF
the sum a
b is dened as c where c
a
b
and
c
a
b
for every , .
Denition .. For every a,
bF
we write a
b if a
b
and a
b
for every
,.
. Fuzzy number valued Lebesgue outer measure
Denition .. Let R , be a fuzzy subset of the real line. The fuzzy number
valued Lebesgue outer measure for the fuzzy subset is dened as m
,
K where
K inf
n
sup
xI
n
x
lI
n
and the inmum is taken over all countable collection I
n
of
open intervals covering R.
Proposition .. If
, then m
m
.
Proof. m
,
K
where K
inf
n
sup
xI
n
x
lI
n
,
K
where
K
inf
n
sup
xI
n
x
lI
n
m
.
Lemma .. If
and
are any two fuzzy sets and
, then
m
m
m
.
Proof. If m
, or m
, then the lemma is trivial.
Suppose that m
, and m
, . Let K
inf
n
sup
xI
n
x
lI
n
and K
inf
n
sup
xI
n
x
lI
n
Choosing gt we can nd countable covers I
,I
, and I
,I
, of open intervals
for R such that
n
sup
xI
n
x
lI
n
lt K
/ and
n
sup
xI
n
x
lI
n
lt K
/.
Set another sequence J
,J
, of pairwise disjoint open intervals such that R
n
J
n
T
say is countable and such that each J
n
is contained in some I
i
and I
j
. Choose a sequence of
Vol. On fuzzy number valued Lebesgue outer measure
open intervals J
,J
, such that
n
lJ
n
lt / and T
n
J
n
.
Setting I
,I
,J
,J
,J
,J
we can nd I
,I
, is a cover of R.
Also
n
sup
xI
n
x
lI
n
n
sup
xJ
n
x
lJ
n
n
sup
xJ
n
x
lJ
n
n
sup
xI
n
x
lI
n
n
lJ
n
K
//K
/.
Similarly we nd
n
sup
xI
n
x
lI
n
K
/.
If K inf
n
sup
xI
n
x
lI
n
, then
K
n
sup
xI
n
x
lI
n
n
sup
xI
n
x
lI
n
n
sup
xI
n
x
lI
n
K
/K
/
and hence K K
K
.
Therefore
m
,
K,
K
K
,
K
,
K
m
m
Remark .. By using the induction on n the result of above lemma can be extended as
m
m
m
m
n
whenever
n
.
Lemma .. m
,
K where K inf
n
sup
xA
n
x
mA
n
and the inmum is
taken over all sequences A
n
of Lebesgue measurable subsets of R such that R
n
A
n
.
Proof. Let K inf
n
sup
xI
n
x
lI
n
and K
inf
n
sup
xA
n
x
mA
n
We have to prove that K K
for which it is enough to show that K K
.
Suppose that K
lt . Choosing gt , we can nd a sequence
n
with
n
gt for all n
such that
n
n
lt /. Accordingly we can have a sequence of Lebesgue measurable subsets of
R such that R
n
A
n
and such that
n
sup
xA
n
x
mA
n
lt K
/.
T. Veluchamy and P. S. Sivakkumar No.
To each n, nd a sequence of pairwise disjoint open intervals such that m
k
I
n
k
A
n
lt
n
and such that
k
I
n
k
A
n
M
n
say is countable.
Finding a sequence J
n
of open intervals such that
n
M
n
n
J
n
and such that
n
lJ
n
lt /.
Clearly I
n
k
,J
n
k , , , n , , is a cover for R and we have
n
k
sup
xI
n
k
x
lI
n
k
n
sup
xJ
n
x
lJ
n
n
sup
xA
n
x
mA
n
n
n
lJ
n
K
n
n
/K
as x , m
k
I
n
k
A
n
lt
n
and I
n
k
, k , are pairwise disjoint for
every n.
Therefore K K
. and hence the result follows.
Theorem .. If E is a Lebesgue measurable subset of the real line and is a fuzzy subset
of R then m
m
Em
E
c
.
Proof.
m
m
EE
c
m
EE
c
m
Em
E
c
.
Suppose that m
,
K, m
E,
K
and m
E
c
,
K
.
If K , the result is trivial.
Suppose that K . Choosing gt we can nd a sequence A
n
of pairwise disjoint
measurable subsets of R such that
n
A
n
R and
n
sup
xA
n
x
mA
n
lt K .
Vol. On fuzzy number valued Lebesgue outer measure
Therefore by previous lemma,
K
K
n
sup
xA
n
E
x
E
x
mA
n
E
n
sup
xA
n
E
c
x
E
x
mA
n
E
c
n
sup
xA
n
E
x
E
cx
mA
n
E
n
sup
xA
n
E
c
x
E
cx
mA
n
E
c
n
sup
xA
n
E
x
mA
n
E
n
sup
xA
n
E
c
x
mA
n
E
c
n
sup
xA
n
x
mA
n
K.
Therefore
m
Em
E
c
,
K
,
K
,
K
K
,
Km
.
Hence the result.
Remark .. Using above theorem we can dene Lebesgue measurable fuzzy subset as
follows
A fuzzy subset is Lebesgue measurable i m
m
m
c
for some
complement
c
of .
. General Fuzzy number valued Lebesgue outer measure
We shall assume X as a nonempty set, denotes a algebra of subsets of X and m
denotes a positive measure on .
Denition .. Let X , be a fuzzy subset of X. The fuzzy number valued outer
measure for the fuzzy subset is dened as m
,
K where K inf
n
sup
xA
n
x
mA
n
and the inmum is taken over all countable collection A
n
in which cover X.
The following result can be proved that is analogous to lemma ..
If
,
,
n
are fuzzy subsets of X and if
n
then m
m
m
m
n
.
Let
EXm
Am
AEm
AE
c
for every A X. Then
is
algebra,
. If mA m
A for every A
then m is a positive measure on
. If
m
is nite nite then m is nite nite. We use mA instead of m
A whenever
A
.
T. Veluchamy and P. S. Sivakkumar No.
Lemma .. If is fuzzy set on X then m
,
K, where
K inf
n
sup
xA
n
x
mA
n
and the inmum is taken over all sequences A
n
of sets in
such that X
n
A
n
.
Proof. The proof is similar to that of lemma . with the following modications.
To each n, such that sup
xA
n
x , choose a sequence such that A
n
k
A
n
k
,A
n
k
,A
n
k
are pairwise disjoint, and
m
k
A
n
k
A
n
lt
n
.
Let B be the set obtained from X after removing all A
n
k
. Then
A
n
k
k , , B is a cover for X which leads to the conclusion of the as in lemma
..
Theorem .. If E
and is a fuzzy subset of X , then
m
m
Em
E
c
Proof. Analogous to the proof in lemma .
Remark .. Using above theorem if we wish to dene measurable fuzzy subset of X
as those which satises m
m
m
c
for some complement
c
of then
must be
measurable.
References
Z.Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, .
Vimala, Fuzzy probability via fuzzy measures, Ph.D Dissertation, Karaikudi Alagappa
University, .
Minghu Ha and Ruisheng Wang, Outer measures and inner measures on fuzzy measure
spaces, The journal of Fuzzy Mathematics, , .
Arnold Kaufmann, Madan M.Gupta, Introduction to Fuzzy Arithmetic Theory and
Applications, International Thomson Computer Press, .
Scientia Magna
Vol. , No. ,
A note to Lagrange mean value Theorem
Dewang Cui, Wansheng He and Hongming Xia
Department of Mathematics, Tianshui Normal University,
Tianshui, Gansu , P.R.China
Abstract In this paper we prove that the intermediate point of Lagrange mean value
theorem is a function, furthermore, we study its monotonicity, continuity and derivable prop
erty. For application, we give an example to show the incorrect for a proof method in course
of calculus teaching.
Keywords Lagrange mean value theorem, intermediate point , monotonicity, continu
ity, derivable property
. Introduction
Lagrange mean value theorem is one of most important theorems in calculus. It is an
important tool to study the property of function. It sets up a bridge between function and
derivable function. But Lagrange mean value theorem only arms existence of intermediate
point , not states its other properties. Recently, some people have studied asymptotic
qualities of and obtained good results. In this paper, on the base of summarizing related
results, we study the monotonicity, continuity and derivable property of .
. Several lemmas
Lemma . Lagrange mean value theorem Assume that fx is continuous on the closed
interval a, b and derivable on the open interval a, b. Then for x a, b, there at least
exists a point a, x, such that
f
fx fa
xa
,
In reference, they obtain some important results, among them the most typical result
is as follows.
Lemma . Assume that fx is a rstorder continuous and derivable function on interval
a, b, and f
xf
a is order innitesimal of x a, where gt . Then with has
asymptotic estimator
lim
ba
a
ba
a
a
.
This work is supported by the Gansu Provincial Education Department Foundation .
Dewang Cui, Wansheng He and Hongming Xia No.
. Main results and their proofs
Assume that fx satises the conditions of lemma when x a, b, then for x a, b,
when a is xed, the intermediate point changes with x and has important properties as
follows.
Theorem. Assume that fx is continuous on the closed interval a, b and derivable on
the open interval a, b, and fx is secondorder continuous and derivable in interval a, b,
f
x is strictly monotone in intervala, b , f
x is always positive or always negative in
interval a, b, then we have
i The point with is a uniform function about x, notes x
ii x is a monotone increasing function about x
iii x is a continuous function about x
iv x is a derivable function about x and
x
f
xf
x
x af
x
.
Proof. i Because f
x is strictly monotone in interval a, b, we can easily prove that
i holds.
ii Assume that f
x is monotone increasing in interval a, b, for x
,x
a, b and
x
lt x
, by given condition and lemma we have
fx
fa f
x
x
a, fx
fa f
x
x
a.
So fx
fx
f
x
f
x
x
af
x
x
x
. Also by fx
fx
f
x
x
, hence
f
f
x
x
x
f
x
f
x
x
a,
where x
lt lt x
, a lt x
lt x
, a lt x
lt x
, a lt x
lt x
lt b.Because f
x is
monotone increasing, we have f
gt f
x
, and
f
f
x
x
af
x
x
f
x
af
x
x
a
gt f
x
x
f
x
x
af
x
x
a.
But f
x
x
fx
fx
,
f
x
x
a fx
fa,
f
x
x
a fx
fa,
So f
f
x
x
a gt ,
f
f
x
gt ,
f
x
f
x
gt .
By monotone increasing f
x, we have
x
gt x
.
When f
x is monotone decreasing in interval a, b, the proof is same to above proof.
iii By given conditions and Lemma , we have
f
x
fx fa
xa
,
and
f
xh
fx h fa
xha
,
so
Vol. A Note to Lagrange Mean Value Theorem
f
xhf
x
x afx h fa hfx fa
x h ax a
.
By ii, we know that x is a monotone function about x. When h , also by Lemma
, we have
f
xhf
xf
x h x,
where lies between x h and x, so
lim
h
x h x lim
h
x afx h fa hfx fa
f
x h ax a
.
That is to say, x is continuous in interval a, b.
iv By denition of derivative, we have
x lim
h
xhx
h
lim
h
xa
fx h fx
h
fx fa
f
x h ax a
x af
x fx fa
f
xx a
f
xf
x
x af
x
.
Thus, the proof is complete.
Finally, it deserves to mention that, in the course of teaching higher mathematics, some
people consider the application of this theorem as an incorrect proof method. In fact, this
thinking is not right. we give an example.
Example. Assume that fx is quadratic dierentiable on the closed interval a, b and
f
x gt , we try to prove that the function
gx
fx fa
xa
is a monotone increasing function in the open interval a, b. Proof By Lemma, we have
fx fa f
xx a, a lt x lt x b.
So
gx
fx fa
xa
f
x.
By above theorem we know that x is a monotone increasing and derivable function in
the open interval a, b, also by derivative rules of compound function, we have
g
xf
x
x,
but f
x gt and
x gt , so
g
xf
x
x gt .
Thus, the proof is complete.
Dewang Cui, Wansheng He and Hongming Xia No.
References
Wang Zewen, etc, the Inverse Problem to Mean Value Theorem of Dierentials and Its
Asymptotic Property, Journal of East China Geological Institute, , No., .
Li Wenrong, Asymptotic Property of intermediate point to Mean Value Theorem of
Dierentials, Mathematics in Practice and Theory, , No., .
Zhang Guangfan, A Note on Mean Value Theorem of Dierentials, Mathematics in
Practice and Theory, , No., .
Dai Lihui, etc, the Trend of Change of to Mean Value Theorem of Dierentials, Chinese
Journal of Engineering Mathematics, , No., .
Wu Liangsen, etc, Mathematics Analysis, East China Normal University Publisher,
, No., .
Scientia Magna
Vol. , No. ,
Some properties of , fuzzy BGalgebras
L. Torkzadeh
and A. Boruman Saeid
Department of Math., Islamic Azad University, Kerman branch, Kerman, Iran
Department of Math., Shahid Bahonar University of Kerman, Kerman, Iran
ltorkzadehyahoo.com arshamiauk.ac.ir
Abstract By two relations belonging to and quasicoincidence q between fuzzy points
and fuzzy sets, we dene the concept of , fuzzy subalgebras where , are any two of
, q, q, q with q. We state and prove some theorems in , fuzzy BG
algebras.
Keywords BGalgebra, , fuzzy subalgebra, fuzzy point.
. Introduction
Y. Imai and K. Iseki introduced two classes of abstract algebras BCKalgebras and
BCIalgebras. It is known that the class of BCKalgebras is a proper subclass of the class
of BCIalgebras. In J. Neggers and H. S. Kim introduced the notion of dalgebras, which
is generalization of BCKalgebras and investigated relation between dalgebras and BCK
algebras. Also they introduced the notion of Balgebras . In C. B. Kim, H. S. Kim
introduced the notion of BGalgebras which is a generalization of Balgebras. In , P. M.
Pu and Y. M. Liu , introduced the idea of quasicoincidence of a fuzzy point with a fuzzy set,
which is used to generate some dierent types of fuzzy subgroups, called , fuzzy subgroups,
introduced by Bhakat and Das . In particular, , qfuzzy subgroup is an important
and useful generalization of Rosenfelds fuzzy subgroup. In this note we introduced the notion
of , fuzzy BGalgebras. We state and prove some theorems discussed in , fuzzy BG
subalgebras and level subalgebras.
. Preliminary
Denition .. A BGalgebra is a nonempty set X with a consonant and a binary
operation satisfying the following axioms
Ixx,
II x x,
III x y y x, for all x, y X.
For brevity we also call X a BGalgebra. In X we can dene a binary relation by x y
if and only if x y .
Theorem .. In a BGalgebra X, we have the following properties
L. Torkzadeh and A. Borumand Saeid No.
Theorem .. In a BGalgebra X, we have the following properties
For all x, y X,
i x x,
ii if x y , then x y,
iii if x y, then x y,
iv x x x x.
Theorem .. A nonempty subset I of a BGalgebra X is called a subalgebra of X
if x y I for any x, y I.
A mapping f X Y of BGalgebras is called a BGhomomorphism if fx y
fx fy for all x, y X.
We now review some fuzzy logic concepts see and .
Let X be a set. A fuzzy set A on X is characterized by a membership function
A
X
,.
Let f X Y be a function and B a fuzzy set of Y with membership function
B
. The
inverse image of B, denoted by f
B, is the fuzzy set of X with membership function
f
B
dened by
f
B
x
B
fx for all x X.
Conversely, let A be a fuzzy set of X with membership function
A
. Then the image of A,
denoted by fA, is the fuzzy set of Y such that
fA
y
sup
xf
y
A
x if f
y,
otherwise.
A fuzzy set of a set X of the form
y
t if y x,
otherwise.
where t , is called a fuzzy point with support x and value t and is denoted by x
t
.
Consider a fuzzy point x
t
, a fuzzy set on a set X and , q, q, q, we dene
x
t
as follow
ix
t
resp. x
t
q means that x t resp. x t gt and in this case we said
that x
t
belong to resp. quasicoincident with fuzzy set .
ii x
t
q resp. x
t
q means that x
t
or x
t
q resp. x
t
and x
t
q.
Denition .. Let be a fuzzy set of a BGalgebra X. Then is called a fuzzy
BGalgebra subalgebra of X if
x y minx, y
for all x, y X.
Example .. Let X , , , be a set with the following table
Vol. Some properties of , fuzzy BGalgebras
Then X, , is a BGalgebra. Dene a fuzzy set X , on X, by t
and t
, for t
,t
, and t
gt t
. Then is a fuzzy BGalgebra of X.
Denition .. Let be a fuzzy set of X. Then the upper level set U of X is
dened as following
UxXx.
Denition .. Let f X Y be a function. A fuzzy set of X is said to be
finvariant, if fx fy implies that x y, for all x, y X.
. , fuzzy BGalgebras
From now on X is a BGalgebra and , , q, q, q unless otherwise specied.
By x
t
we mean that x
t
does not hold.
Theorem .. Let be a fuzzy set of X. Then is a fuzzy BGalgebra if and only if
x
t
,y
t
xy
mint
,t
,
for all x, y X and t
,t
,.
Proof. Assume that is a fuzzy BGalgebra. Let x, y X and x
t
,y
t
, for t
,t
, . Then x t
and y t
, by hypothesis we can conclude that
x y minx, y mint
,t
.
Hence x y
mint
,t
.
Conversely, Since x
x
and y
y
for all x, y X, then x y
minx,y
.
Therefore x y minx, y.
Note that if is a fuzzy set of X dened by x . for all x X, then the set
x
t
x
t
q is empty.
Denition .. A fuzzy set of X is said to be an , fuzzy subalgebra of X, where
q, if it satises the following condition
x
t
,y
t
xy
mint
,t
for all t
,t
,.
Proposition .. is an , fuzzy subalgebra of X if and only if for all t , , the
nonempty level set U t is a subalgebra of X.
Proof. The proof follows from Theorem ..
L. Torkzadeh and A. Borumand Saeid No.
Example .. Let X , , , be a set with the following table
Then X, , is a BGalgebra. Let be a fuzzy set in X dened ., . and
.. Then is an , qfuzzy subalgebra of X. But
is not an , fuzzy subalgebra of X since
.
and
.
, but
min.,.
.
.
is not a q, qfuzzy subalgebra of X since
.
q and
.
q, but
min.,.
.
q.
is not an q, qfuzzy subalgebra of X since
.
q and
.
q, but
min.,.
.
q.
Theorem .. Let be a fuzzy set. Then the following diagram shows the relationship
between , fuzzy subalgebras of X, where , are one of and q.
,
,
d
d
d
d
ds
,
d
d
d
d
ds
,
and also we have
q, q
q,
d
d
d
d
ds
q, q
d
d
d
d
ds
q, q
Proof. The proof is easy.
Proposition .. If is a nonzero , fuzzy subalgebra of X, then gt .
Proof. Assume that . Since is nonzero, then there exists x X such that
Vol. Some properties of , fuzzy BGalgebras
x t gt . Thus x
t
for or q, but x x
mint,t
t
. This is a
contradiction. Also x
where q, since x t gt . But x x
min,
,
which is a contradiction. Hence gt .
For a fuzzy set in X, we denote the support by, X
x X x gt .
Proposition .. If is a nonzero , qfuzzy subalgebra of X, then the set X
is a
subalgebra of X.
Proof. Let x, y X
. Then x gt and y gt . Suppose that x y , then
x
x
and y
y
, but x y lt minx, y and x y minx, y ,
i.e x y
minx,y
q, which is a contradiction . Hence x y X
. Therefore X
is a
subalgebra of X.
Proposition .. If is a nonzero q, qfuzzy subalgebra of X, then the set X
is a
subalgebra of X.
Proof. Let x, y X
. Then x gt and y gt . Thus x gt and y gt
imply that x
q and y
q. If xy , then xy lt min, and xymin,
. Thus x y
min,
q, which is a contradiction. It follows that x y gt and so
xyX
.
Theorem .. Let be a nonempty , fuzzy subalgebra, where , , q, q,
q and q. Then X
is a subalgebra of X.
Proof. The proof follows from Theorem . and Propositions . and ..
Theorem .. Any nonzero q, qfuzzy subalgebra of X is constant on X
.
Proof. Let be a nonzero q, qfuzzy subalgebra of X. On the contrary, assume that
is not constant on X
. Then there exists y X
such that t
y
yt
. Suppose
that t
y
lt t
and so t
lt t
y
lt . Thus there exists t
,t
, such that t
lt t
lt
t
y
lt t
lt . Then t
t
t
gt and y t
t
y
t
gt . So
t
q and y
t
q.
Since
y mint
,t
yt
t
y
t
lt ,
we get that y
mint
,t
q, which is a contradiction. Now let t
y
gt t
and t
. Then
yt
t
y
t
gt , i.e y
t
q. Since
yyt
t
t
t
,
then we get that y y
mint
,t
q, which is a contradiction. Therefore is constant on
X
.
Theorem .. is a nonzero q, qfuzzy subalgebra if and only if there exists subal
gebra S of X such that
x
t if x S
otherwise
for some t , .
Proof. Let be a nonzero q, qfuzzy subalgebra. Then by Proposition . and Theorems
L. Torkzadeh and A. Borumand Saeid No.
. and . we have gt , X
is a subalgebra of X and
x
if x X
otherwise
Conversely, let x
t
q and y
t
q, for t
,t
, . Then x t
gt and y t
gt imply
that x and y . Thus x, y S and so x y S. Hence x y mint
,t
t mint
,t
gt . Therefore is a q, qfuzzy subalgebra of X.
Theorem .. is a nonzero q, qfuzzy subalgebra of X if and only if U X
and for all t , , the nonempty level set U t is a fuzzy subalgebra of X.
Proof. Let be a nonzero q, qfuzzy subalgebra. Then by Theorem . we have
x
if x X
otherwise
So it is easy to check that U X
. Let x, y U t, for t , . Then x t and
y t. If t , then it is clear that x y U . Now let t , . Then x, y X
and
so x y X
. Hence x y t. Therefore U t is a subalgebra of X.
Conversely, since U X
and U , then X
is a subalgebra of X. Also
UX
and X imply that is nonzero. Now let x X
. Then x and
x gt . Since U x , so U x is a subalgebra of X. Then U x imply
that x. Hence x , for all x X
i.e
x
if x X
otherwise
Therefore by Theorem . is a q, qfuzzy subalgebra of X.
Example .. Let X , , , be BGalgebra in Example .. Dene fuzzy set on
X by
., ..
Then X
X, U X
and also
Ut
X if t .
if . lt t .
if t gt .
is a subalgebra of X, while by Theorem ., is not a q, qfuzzy subalgebra.
Theorem .. Every q, qfuzzy subalgebra is an , fuzzy subalgebra.
Proof. The proof follows from Theorem . and Proposition ..
Note that in Example . is an , fuzzy subalgebra, while it is not a q, qfuzzy
subalgebra. So the converse of the above theorem is not true in general.
Theorem .. If is a nonzero fuzzy set of X. Then there exists subalgebra S of X
such that
S
if and only if is an , fuzzy subalgebra of X, where , is one of the
following forms
Vol. Some properties of , fuzzy BGalgebras
i , q, ii , q,
iii q, , iv q, q,
v q, q, vi q, q,
vii q, .
Proof. Let
S
. We show that is , qfuzzy subalgebra. Let x
t
and
x
t
, for t
,t
, . Then x t
and y t
imply that x, y S. Thus
x y S, i.e x y . Therefore x y mint
,t
and x y mint
,t
gt ,
i.e x y
mint
,t
q. Similar to above argument, we can see that is an , fuzzy
subalgebra of X, where , is one of the above forms.
Conversely, we show that
X
. Suppose that there exists x X
such that x lt . Let
, choose t , such that t lt min x, x, . Then x
t
and
t
, but
x
mint,t
x
t
, where q or q. Which is a contradiction. If q, then
x
and
, while x
min,
x
where or q, which is a contradiction.
Now let q and choose t , such that x
t
but x
t
q. Then x
t
and
but
x
mint,
x
t
for q or q, which is a contradiction. Finally we have x
q
and
q but x
min,
x
, which is a contradiction. Therefore
X
.
Theorem .. Let S be a subalgebra of X and let be a fuzzy set of X such that
a x for all x XS,
b x . for all x S.
Then is a q, qfuzzy subalgebra of X.
Proof. Let x, y X and t
,t
, be such that x
t
q and y
t
q. Then we get that
xt
gt and y t
gt . We can conclude that x y S, since in otherwise x XS
or y XS and therefore t
gt or t
gt which is a contradiction. If mint
,t
gt ., then
xymint
,t
gt and so xy
mint
,t
q. If mint
,t
., then xy mint
,t
and thus x y
mint
,t
. Hence x y
mint
,t
q.
Theorem .. Let be a q, qfuzzy subalgebra of X such that is not constant
on the set X
. Then there exists x X such that x .. Moreover, x . for all
xX
.
Proof. Assume that x lt . for all x X. Since is not constant on X
, then
there exists x X
such that t
x
xt
. Let t
lt t
x
. Choose gt . such
that t
lt lt t
x
. It follows that x
q, x x t
lt min, and
xx min, t
lt . Thus xx
min,
q, which is a contradiction.
Now, if t
x
lt t
then we can choose gt . such that t
x
lt lt t
. Thus
q and
x
q, but x
min,
x
q, because x lt . lt and x t
x
lt , which
is a contradiction. Hence x . for some x X. Now we show that .. On the
contrary, assume that t
lt .. Since there exists x X such that x t
x
., it
follows that t
lt t
x
. Choose t
gt t
such that t
t
lt lt t
x
t
. Then xt
t
x
t
gt ,
and so x
t
q. Thus we can conclude that
x x mint
,t
t
t
t
lt ,
L. Torkzadeh and A. Borumand Saeid No.
and
xxt
lt t
mint
,t
.
Therefore x x
mint
,t
q, which is a contradiction. Thus .. Finally we prove
that x . for all x X
. On the contrary, let x X
and t
x
x lt .. Consider
lt t lt . such that t
x
t lt .. Then x t
x
gt and . t gt , imply
that x
q and
.t
q. But x
min,.t
x
.t
q, since x x lt . t
and x .t t
x
.t lt .. . Which is a contradiction. Therefore x .
for all x X
.
Theorem .. Let be a nonzero fuzzy set of X. Then is a q, qfuzzy subalgebra
of X if and only if there exists subalgebra S of X such that
x
a if x S
otherwise
or x
. if x S
otherwise
for some a , .
Proof. Let be a q, qfuzzy subalgebra of X. If is constant on X
, then x
if x X
otherwise
. If is not constant on X
, then by Theorem . we have x
. if x X
otherwise
. Conversely, the proof follows from Theorems ., . and ..
Theorem .. Let be a nonzero q, qfuzzy subalgebra of X. Then the nonempty
level set U t is a subalgebra of X, for all t , ..
Proof. If is constant on X
, then by Theorem ., is a q, qfuzzy subalgebra. Thus
by Theorem . we have the nonempty level set U t is a subalgebra of X, for t , .
If is not constant on X
, then by Theorem ., we have x
. if x X
otherwise
.
Now we show that the nonempty level set U t is a subalgebra of X for t , .. If t ,
then it is clear that U t is a subalgebra of X. Now let t , . and x, y U t. Then
x, y t gt imply that x, y X
. Thus x y X
and so x y . t. Therefore
x y U t.
Theorem .. Let be a nonzero fuzzy set of X, U . X
and the nonempty
level set U t is a subalgebra of X, for all t , . Then is a q, qfuzzy subalgebra
of X.
Proof. Since we get that X
. Thus by hypothesis we have U . and
so X
is a subalgebra of X. Also x ., for all x X
and x , if x X
. Therefore
by Theorem ., is a q, qfuzzy subalgebra of X.
Theorem .. A fuzzy set of X is an , qfuzzy subalgebra of X if and only if
x y minx, y, ., for all x, y X.
Proof. Let be an , qfuzzy subalgebra of X and x, y X. If x or y ,
then x y minx, y, .. Now let x and y . If minx, y lt ., then
x y minx, y. Since, assume that x y lt minx, y, then there exists
Vol. Some properties of , fuzzy BGalgebras
t gt such that x y lt t lt minx, y. Thus x
t
and y
t
but x y
mint,t
xy
t
q, since xy lt t and xy t lt lt t lt , which is a contradiction. Hence if
minx, y lt ., then x y minx, y. If minx, y ., then x
.
and y
.
. So we can get that
xy
min.,.
xy
.
q.
Then x y gt .. Consequently, x y minx, y, ., for all x, y X.
Conversely, let x, y X and t
,t
, be such that x
t
and y
t
. So x t
and y t
. Then by hypothesis we have x y minx, y, . mint
,t
, ..
If mint
,t
., then x y minx, y. If mint
,t
gt ., then x y ..
Thus x y mint
,t
gt . Therefore x y
mint
,t
q.
Theorem .. Let be an , qfuzzy subalgebra of X.
i If there exists x X such that x ., then .
ii If lt ., then is an , fuzzy subalgebra of X.
Proof. i Let x .. Then by hypothesis we have xx minx, x, .
..
ii Let lt .. Then by i x lt ., for all x X. Now let x
t
and y
t
,
for t
,t
, . Then x t
and y t
. Thus x y minx, y, .
mint
,t
, . mint
,t
. Therefore x y
mint
,t
.
Lemma .. Let be a nonzero , q fuzzy subalgebra of X. Let x, y X such
that x lt y. Then
xy
x if y lt . or x lt . y
. if x .
.
Proof. Let y lt .. Then we have x y minx, y, . x. Also
x x y y minx y, y, .
Now we show that y y. Since y lt ., then yy miny, y, .
y. Thus y min, y, . y. Hence and hypothesis imply that
x minxy, y. Since x lt y, then x xy. Therefore xy x.
Now let x lt . y. Then similar to above argument x y x and x
minx y, y, .. Since y ., then by Theorem .i, .. Thus
y min, y, . .. So by hypothesis we get that x minx y, ..
Thus x lt . imply that x x y. Therefore x y x. Let x .. Then
x y minx, y, . ..
Theorem .. Let be an , qfuzzy subalgebra of X. Then for all t , .,
the nonempty level set U t is a subalgebra of X. Conversely, if the nonempty level set is
a subalgebra of X, for all t , , then is an , qfuzzy subalgebra of X.
Proof. Let be an , qfuzzy subalgebra of X. If t , then U t is a subalgebra
of X. Now let U t , lt t . and x, y U t. Then x, y t. Thus by
hypothesis we have x y minx, y, . mint, . t. Therefore U t is a
subalgebra of X.
L. Torkzadeh and A. Borumand Saeid No.
Conversely, let x, y X. Then we have
x, y minx, y, . t
.
Hence x, y U t
, for t
, and so x y U t
. Therefore x y t
minx, y, ., i.e is an , qfuzzy subalgebra of X.
Theorem .. Let S be a subset of X. The characteristic function
S
of S is an
, qfuzzy subalgebra of X if and only if S is a subalgebra of X.
Proof. Let X
S
be an , qfuzzy subalgebra of X and x, y S. Then
S
x
S
y, and so x
S
and y
S
. Hence x y
xy
min,
q
S
, which implies that
S
x y gt . Thus x y S. Therefore S is a subalgebra of X.
Conversely, if S is a subalgebra of X, then
S
is an , fuzzy subalgebra of X. So by
Theorem . we get that is an , qfuzzy subalgebra of X.
Lemma .. Let f X Y be a BGhomomorphism and G be a fuzzy set of Y with
membership function
G
. Then x
t
f
G
fx
t
G
, for all , q, q, q.
Proof. Let . Then
x
t
f
G
f
G
xt
G
fx t fx
t
G
.
The proof of the other cases is similar to above argument.
Theorem .. Let f X Y be a BGhomomorphism and G be a fuzzy set of Y with
membership function
G
.
i If G is an , fuzzy subalgebra of Y , then f
G is an , fuzzy subalgebra of X,
ii Let f be epimorphism. If f
G is an , fuzzy subalgebra of X, then G is an
, fuzzy subalgebra of Y .
Proof. i Let x
t
f
G
and y
r
f
G
, for t, r , . Then by Lemma ., we
get that fx
t
G
and fy
r
G
. Hence by hypothesis fx fy
mint,r
G
. Then
fx y
mint,r
G
and so x y
mint,r
f
G
.
ii Let x, y Y . Then by hypothesis there exist x
,y
X such that fx
x and fy
y. Assume that x
t
G
and y
r
G
, then fx
t
G
and fy
r
G
. Thus x
t
f
G
and
y
r
f
G
and therefore x
y
mint,r
f
G
. So
fx
y
mint,r
G
fx
fy
mint,r
G
xy
mint,r
G
.
Theorem .. Let f X Y be a BGhomomorphism and H be an , qfuzzy
subalgebra of X with membership function
H
. If
H
is an finvariant, then fH is an
, qfuzzy subalgebra of Y .
Proof. Let y
and y
Y . If f
y
or f
y
, then
fH
y
y
min
fH
y
,
fH
y
, ..
Now let f
y
and f
y
. Then there exist x
,x
X such that fx
y
and
fx
y
. Thus by hypothesis we have
fH
y
y
sup
tf
y
y
H
t
sup
tf
fx
x
H
t
Vol. Some properties of , fuzzy BGalgebras
H
x
x
since
H
is an finvariant
min
H
x
,
H
x
,.
min sup
tf
y
H
t, sup
tf
y
H
t, .
min
fH
y
,
fH
y
, ..
So by Theorem ., fH is an , qfuzzy subalgebra of Y .
Lemma .. Let f X Y be a BGhomomorphism.
i If S is a subalgebra of X, then fS is a subalgebra of Y
ii If S
is a subalgebra of Y , then f
S
is a subalgebra of X.
Proof. The proof is easy.
Theorem .. Let f X Y be a BGhomomorphism . If H is a nonzero q, qfuzzy
subalgebra of X with membership function
H
, then fH is a nonzero q, qfuzzy subalgebra
of Y .
Proof. Let H be a nonzero q, qfuzzy subalgebra of X. Then by Theorem ., we have
H
x
H
if x X
otherwise
. Now we show that
fH
y
H
if y fX
otherwise
.
Let y Y . If y fX
, then there exist x X
such that fx y. Thus
fH
y
sup
tf
y
H
t
H
. If y fX
, then it is clear that
fH
y . Since X
is subal
gebra of X, then fX
is a subalgebra of Y . Therefore by Theorem ., fH is a nonzero
q, qfuzzy subalgebra of Y .
Theorem .. Let f X Y be a BGhomomorphism . If H is an , fuzzy subal
gebra of X with membership function
H
, then fH is an , fuzzy subalgebra of Y , where
, is one of the following form
i , q, ii , q,
iii q, , iv q, q,
v q, q, vi q, q,
vii q, , viii q, q.
Proof. The proof is similar to the proof of Theorem ., by using of Theorems . and
..
Theorem .. Let f X Y be a BGhomomorphism and H be an , fuzzy
subalgebra of X with membership function
H
. If
H
is an finvariant, then fH is an
, fuzzy subalgebra of Y .
Proof. Let z
t
fH
and y
r
fH
, where t, r , . Then
fH
z t and
fH
y r. Thus f
z, f
y imply that there exists x
,x
X such that fx
z
and fx
y. since
H
is an finvariant, then
fH
z t and
fH
y r imply that
H
x
t and
H
x
r. So by hypothesis we have
L. Torkzadeh and A. Borumand Saeid No.
fH
z y sup
tf
zy
H
t
sup
tf
fx
x
H
t
H
x
x
mint, r.
Therefore z y
mint,r
fH
, i.e fH is an , fuzzy subalgebra of Y .
Theorem .. Let
i
i be a family of , qfuzzy subalgebra of X. Then
i
i
is an , qfuzzy subalgebra of X.
Proof. By Theorem . we have, for all i
i
x y min
i
x,
i
y, ..
Therefore x y inf
i
i
x y inf
i
min
i
x,
i
y, .
mininf
i
i
x, inf
i
i
y, .
minx, y, ..
Therefore by Theorem ., is an , qfuzzy subalgebra.
Theorem .. Let
i
i be a family of , fuzzy subalgebra of X. Then
i
i
is an , fuzzy subalgebra of X.
Proof. Let x
t
and y
r
, t, r , . Then x t and y r. Thus for all
i,
i
x t and
i
y r imply that
i
x y mint, r. Therefore x y mint, r
i.e x y
mint,r
.
Theorem .. Let
i
i be a family of , fuzzy subalgebra of X. Then
i
i
is an , fuzzy subalgebra of X, where , is one of the following form
i , q, ii , q,
iii q, , iv q, q,
v q, q, vi q, q,
vii q, , viii q, q,
ix q, q.
Proof. We prove theorem for q, qfuzzy subalgebra. The proof of the other cases is
similar, by using Theorems . and ..
If there exists i such that
i
, then . So is a q, qfuzzy subalgebra. Let
i
for all i . Then by Theorem . we have
i
x
i
if x X
i
otherwise
, for all
Vol. Some properties of , fuzzy BGalgebras
i . So it is clear that x
if x
i
X
i
otherwise
. Since
i
X
i
is a subalgebra of X,
then by Theorem . is a q, qfuzzy subalgebra of X.
References
S. S. Ahn and H. D. Lee, Fuzzy Subalgebras of BGalgebras, Commun. Korean Math.
Soc., .
S. K Bhakat and P. Das, , qfuzzy subgroups, Fuzzy Sets and Systems, ,
.
Y. Imai and K. Iseki, On axiom systems of propositional calculi, XIV Proc. Japan
Academy, , .
C. B. Kim, H. S. Kim, On BGalgebras, submitted.
J. Meng and Y. B. Jun, BCKalgebras, Kyung Moonsa, Seoul, Korea, .
J. Neggers and H. S. Kim, On Balgebras, Math. Vensik, , .
J. Neggers and H. S. Kim, On dalgebras, Math. Slovaca, , .
P. M. Pu and Y. M. Liu, Fuzzy Topology I, Neighborhood structure of a fuzzy point
and MooreSmith convergence, J. Math. Anal. Appl., , .
A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., , .
L. A. Zadeh, Fuzzy Sets, Inform. Control, , .
Scientia Magna
Vol. , No. ,
On the Smarandache totient function
and the Smarandache power sequence
Yanting Yang and Min Fang
Department of Mathematics, Northwest University, Xian, Shaanxi, P.R.China
Abstract For any positive integer n, let SPn denotes the Smarandache power sequence.
And for any Smarandache sequence an, the Smarandache totient function Stn is dened
as an, where n is the Euler totient function. The main purpose of this paper is
using the elementary and analytic method to study the convergence of the function
S
S
, where
S
n
k
Stk
,S
n
k
Stk
, and give an interesting limit Theorem.
Keywords Smarandache power function, Smarandache totient function, convergence.
. Introduction and results
For any positive integer n, the Smarandache power function SPn is dened as the smallest
positive integer m such that n m
m
, where m and n have the same prime divisors. That is,
SPn min
mnm
m
, m N,
pn
p
pm
p
.
For example, the rst few values of SPn are SP , SP , SP , SP ,
SP , SP , SP , SP , SP , SP , SP ,
SP , SP , SP , SP , . In reference , Professor
F.Smarandache asked us to study the properties of SPn. It is clear that SPn is not a
multiplicative function. For example, SP , SP , SP SP SP.
But for most n, we have SPn
pn
p, where
pn
denotes the product over all dierent prime
divisors of n. If n p
,kp
k
kp
k
, then we have SPn p
k
, where
k . Let n p
p
p
r
r
, for all
i
i , , , r, if
i
p
i
, then
SPn
pn
p.
About other properties of the function SPn, many authors had studied it, and gave some
interesting conclusions. For example, in reference , Zhefeng Xu had studied the mean value
properties of SPn, and obtained a sharper asymptotic formula
nx
SPn
x
p
pp
O
x
,
Vol. On the Smarandache totient function and the Smarandache power sequence
where denotes any xed positive number, and
p
denotes the product over all primes.
On the other hand, similar to the famous Euler totient function n, Professor F.Russo
dened a new arithmetical function called the Smarandache totient function Stn an,
where an is any Smarandache sequence. Then he asked us to study the properties of these
functions. At the same time, he proposed the following
Conjecture. For the Smarandache power sequence SPk,
S
S
converges to zero as
n , where S
n
k
Stk
,S
n
k
Stk
.
In this paper, we shall use the elementary and analytic methods to study this problem,
and prove that the conjecture is correct. That is, we shall prove the following
Theorem. For the Smarandache power function SPk, we have lim
n
S
S
, where
S
n
k
SPk
,S
n
k
SPk
.
. Some lemmas
To complete the proof of the theorem, we need the following two simple Lemmas
Lemma . For any given real number gt , there exists a positive integer N, such
that for all n N, we have n
cn
ln ln n
, where c is a constant.
Proof See reference
Lemma . For the Euler totient fuction n, we have the asymptotic formula
kn
k
ln n AO
ln n
n
,
where A
n
n
nn
n
n ln n
nn
is a constant.
Proof See reference
. Proof of the theorem
In this section, we shall prove our Theorem.
We separate all integer k in the interval , n into two subsets A and B as follows A the
set of all squarefree integers. B the set of other positive integers k such that k , nA. So
we have
kn
SPk
kA
SPk
kB
SPk
.
From the denition of the subset A, we may get
kA
SPk
kA
k
pk
p
k
k
pk
p
.
Yanting Yang and Min Fang No.
By Lemma , we can easily get
k
k
Oln ln k. Note that
kn
k
k
O. And if
k B, then we can write k as k l m, where l is a squarefree integer and m is a squarefull
integer. Let S denote
kB
SPk
, then from the properties of SPk and k we have
S
lmn
l
pm
p
plm
p
mn
pm
p
l
n
m
l
l
l
m
lm
O
ln ln n
mn
pm
p
.
Let Uk
pk
p, then
mn
pm
p
kn
ak
U
k
, where m is a squarefull integer and the
arithmetical function ak is dened as follows
ak
, if k is a squarefull integer
, otherwise.
Note that
ak
U
k
is a multiplicative function. According to the Euler product formula see
reference and , we have
As
k
ak
U
kk
s
p
p
s
p
s
.
From the Perron formulas , for b
ln n
, T , we have
kn
ak
U
k
i
biT
biT
As
n
s
s
ds O
n
b
b
T
O
nmin
,
ln n
T
an
U
n
.
Taking T n, we can get the estimate
O
n
b
b
T
O
nmin
,
ln n
T
an
U
n
Oln n.
Because the function As
n
s
s
is analytic in Re s gt , taking c
ln n
, then we have
i
biT
biT
As
n
s
s
ds
biT
ciT
As
n
s
s
ds
ciT
biT
As
n
s
s
ds
ciT
ciT
As
n
s
s
ds
.
Note that
ciT
ciT
As
n
s
s
ds O
T
T
dy
c
y
Oln n and
biT
ciT
As
n
s
s
ds
O
b
c
n
T
d
O
ln n
. Similarly,
biT
ciT
As
n
s
s
ds O
ln n
. Hence,
kn
ak
U
k
Oln n.
Vol. On the Smarandache totient function and the Smarandache power sequence
So
kn
SPk
Oln n ln ln n
.
Now we come to estimate
kn
SPk
, from the denition of SPn, we may immediately
get that SPn n. Let n p
p
p
s
s
denotes the factorization of n into prime powers,
then SPn p
p
p
s
s
, where
i
. Therefore, we can get that p
p
p
p
p
s
s
p
s
p
p
p
p
p
s
s
p
s
, thus p
p
p
s
s
p
p
p
s
s
. That is, SPn n, according to Lemma , we can easily get
kn
SPk
kn
k
ln n AO
ln n
n
.
Combining and , we obtain
n
k
SPk
n
k
SPk
O
ln n ln ln n
ln n AO
ln n
n
, as n .
This completes the proof of our Theorem.
References
F. Smarandache, Only Problems, Not Solutions, Xiquan Publishing House, Chicago,
.
F. Russo, A set of new Smarandache functions, sequences and conjectures in number
theory, American Research Press, USA, .
Zhefeng Xu, On the mean value of the Smarandache power function, Acta Mathematica
Sinica Chinese series, , No., .
Huanqin Zhou, An innite series involving the Smarandache power function SPn,
Scientia Magna, , No., .
Tom M. Apostol, Introduction to analytical number theory, SpringerVerlag, New York,
.
H. L. Montgomery, Primes in arithmetic progressions, Mich. Math. J., , .
Pan Chengdong and Pan Chengbiao, Foundation of analytic number theory, Science
Press, Beijing, , .
F. Smarandache, Sequences of numbers involved in unsolved problems, Hexis, .
Wenjing Xiong, On a problem of pseudoSmarandachesquarefree function, Journal of
Northwest University, , No., .
Guohui Chen, An equation involving the Euler function, Pure and Applied Mathemat
ics, , No., .
Scientia Magna
Vol. , No. ,
A new additive function and
the F. Smarandache function
Yanchun Guo
Department of Mathematics, Xianyang Normal University
Xianyang, Shaanxi, P.R.China
Abstract For any positive integer n, we dene the arithmetical function Fn as F .
If n gt and n p
p
p
k
k
be the prime power factorization of n, then Fn
p
p
k
p
k
. Let Sn be the Smarandache function. The main purpose of this paper
is using the elementary method and the prime distribution theory to study the mean value
properties of Fn Sn
, and give a sharper asymptotic formula for it.
Keywords Additive function, Smarandache function, Mean square value, Elementary method,
Asymptotic formula.
. Introduction and result
Let fn be an arithmetical function, we call fn as an additive function, if for any positive
integers m, n with m, n , we have fmn fm fn. We call fn as a complete
additive function, if for any positive integers r and s, frs fr fs. In elementary
number theory, there are many arithmetical functions satisfying the additive properties. For
example, if n p
p
p
k
k
denotes the prime power factorization of n, then function n
k
and logarithmic function fn ln n are two complete additive functions,
n k is an additive function, but not a complete additive function. About the properties
of the additive functions, one can nd them in references , and .
In this paper, we dene a new additive function Fn as follows F If n gt and n
p
p
p
k
k
denotes the prime power factorization of n, then Fn
p
p
k
p
k
.
It is clear that this function is a complete additive function. In fact if m p
p
p
k
k
and n p
p
p
k
k
, then we have mn p
p
p
k
k
k
. Therefore, Fmn
p
p
k
k
p
k
Fm Fn. So Fn is a complete additive
function. Now we let Sn be the Smarandache function. That is, Sn denotes the smallest
positive integer m such that n divide m, or Sn minm n m. About the properties
of Sn, many authors had studied it, and obtained a series results, see references , and
. The main purpose of this paper is using the elementary method and the prime distribution
theory to study the mean value properties of Fn Sn
, and give a sharper asymptotic
formula for it. That is, we shall prove the following
Theorem. Let N be any xed positive integer. Then for any real number x gt , we
Vol. A new additive function and the F. Smarandache function
have the asymptotic formula
nx
Fn Sn
N
i
c
i
x
ln
i
x
O
x
ln
N
x
,
where c
i
i , , , N are computable constants, and c
.
. Proof of the theorem
In this section, we use the elementary method and the prime distribution theory to complete
the proof of the theorem. We using the idea in reference . First we dene four sets A, B,
C, D as follows A n, n N, n has only one prime divisor p such that p n and p
n,
p gt n
B n, n N, n has only one prime divisor p such that p
n and p gt n
C n, n N, n has two deferent prime divisors p
and p
such that p
p
n, p
gt p
gt n
D n, n N, any prime divisor p of n satisfying p n
, where N denotes the set of all
positive integers. It is clear that from the denitions of A, B, C and D we have
nx
Fn Sn
nx
nA
Fn Sn
nx
nB
Fn Sn
nx
nC
Fn Sn
nx
nD
Fn Sn
W
W
W
W
.
Now we estimate W
,W
,W
and W
in respectively. Note that Fn is a complete
additive function, and if n A with n pk, then Sn Sp p, and any prime divisor q of
k satisfying q n
, so Fk n
ln n. From the Prime Theorem See Chapter , Theorem
of we know that
x
px
k
i
c
i
x
ln
i
x
O
x
ln
k
x
,
where c
i
i , , , k are computable constants, and c
. By these we have the
estimate
W
nx
nA
Fn Sn
pkx
pkA
Fpk p
pkx
pkA
F
k
k
x
kltp
x
k
pk
ln
pk ln x
k
x
k
kltp
x
k
p
ln x
k
x
k
x
k
ln
x
k
x
ln
x.
Yanchun Guo No.
If n B, then n p
k, and note that Sn Sp
p, we have the estimate
W
nx
nB
Fn Sn
p
kx
pgtk
Fp
kp
kx
kltp
x
k
F
k
kx
kltp
x
k
k
kx
k
x
k
ln x
x
ln x
.
If n D, then Fn n
ln n and Sn n
ln n, so we have
W
nx
nD
Fn Sn
nx
n
ln
nx
ln
x.
Finally, we estimate main term W
. Note that n C, n p
p
k, p
gt p
gt n
gt k. If
k lt p
lt n
, then in this case, the estimate is exact same as in the estimate of W
. If
k lt p
lt p
lt n
, in this case, the estimate is exact same as in the estimate of W
. So by
we have
W
nx
nC
Fn Sn
p
p
kx
p
gtp
gtk
Fp
p
kp
O
x
ln
x
kx
kltp
x
k
p
x
p
k
F
kp
Fk p
O
x
ln
x
kx
kltp
x
k
p
ltp
x
p
k
p
O
kx
kltp
x
k
p
ltp
x
p
k
kp
O
x
ln
x
kx
kltp
x
k
p
N
i
c
i
x
p
k ln
ix
p
k
O
x
p
k ln
N
x
O
x
ln
x
kx
kltp
x
k
p
p
p
O
kx
kltp
x
k
p
ltp
x
p
k
kp
.
Note that
, from the Abels identity See Theorem . of and we have
kx
kltp
x
k
p
pp
kx
kltp
x
k
p
N
i
c
i
p
ln
i
p
O
p
ln
N
p
N
i
kx
kltp
x
k
c
i
p
ln
i
p
O
kx
kltp
x
k
p
ln
N
p
N
i
d
i
x
ln
i
x
O
N
x
ln
N
x
,
Vol. A new additive function and the F. Smarandache function
where d
i
i , , , N are computable constants, and d
.
kx
kltp
x
k
p
ltp
x
p
k
kp
kx
k
p
x
k
p
x
p
k ln x
kx
x
k ln
x
x
ln
x
.
kx
kltp
x
k
p
x
k ln
N
x
kx
x
k
ln
N
x
x
ln
N
x
.
From the Abels identity and we also have the estimate
kx
kltp
x
k
p
x
p
k ln
x
p
k
kx
k
kltp
x
k
xp
ln
x
kp
N
i
b
i
x
ln
i
x
O
x
ln
N
x
,
where b
i
i , , , N are computable constants, and b
.
Now combining , , , , , , and we may immediately deduce the
asymptotic formula
nx
Fn Sn
N
i
a
i
x
ln
i
x
O
x
ln
N
x
,
where a
i
i , , , N are computable constants, and a
b
d
.
This completes the proof of Theorem.
References
C.H.Zhong, A sum related to a class arithmetical functions, Utilitas Math., ,
.
H.N.Shapiro, Introduction to the theory of numbers, John Wiley and Sons, .
Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem in
Chinese, Shanghai Science and Technology Press, Shanghai, .
Xu Zhefeng, On the value distribution of the Smarandache function, Acta Mathematica
Sinica in Chinese, , No., .
Zhang Wenpeng, The elementary number theory in Chinese, Shaanxi Normal Univer
sity Press, Xian, .
Tom M. Apostol. Introduction to Analytic Number Theory, SpringerVerlag, .
Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems in Chinese, High
American Press, .
Chen Guohui, New Progress On Smarandache Problems in Chinese, High American
Press, .
Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress
in Chinese, High American Press, .
SCIENTIA MAGNA
International Book Series