Download Topological Groupoids - Trace: Tennessee Research and Creative

University of Tennessee, Knoxville
Trace: Tennessee Research and Creative
Exchange
Doctoral Dissertations
Graduate School
6-1959
Topological Groupoids
Ronson J. Warne
University of Tennessee - Knoxville
Recommended Citation
Warne, Ronson J., "Topological Groupoids. " PhD diss., University of Tennessee, 1959.
http://trace.tennessee.edu/utk_graddiss/2968
This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been
accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more
information, please contact [email protected].
To the Graduate Council:
I am submitting herewith a dissertation written by Ronson J. Warne entitled "Topological Groupoids." I
have examined the final electronic copy of this dissertation for form and content and recommend that it
be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a
major in Mathematics.
Orville G. Harold, Major Professor
We have read this dissertation and recommend its acceptance:
E. Cohen, Edward D. Harris, T. A. Fisher, D. D. Lillian, W. H. Flecther
Accepted for the Council:
Dixie L. Thompson
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official student records.)
May 27, 1959
To the Graduate Councila
I am submitting herewi th a thesis written by Ronson J. Warne
entitled •Topological Groupoidsn. I re comme nd that it be ac cepted in
part ial fulfillment of the requirements for the degree of Doctor of
Philosophy, wi th a major in Mathematics.
We have read this thesis and
rec ommend its acceptance:
c�
b, b.L-t-.P�
Y,')./.3�
Ac cep ted for the
Council:
'roPOLOOTCAL GROUPOIDS
A THESIS
Submitted to
'lhe Graduate Council
of
The.University of Tennessee
.
Pa rtial
in
Fulfillment of
the Requirements
for the des:ree of
_Doctor of Phil<!>s�hy
Ronson
-
J. Warne
June
.
19S9
ACKNOWLEDGEMI!N T
The author wishe s to acknowledge his indebtedness to Professor
Orville G. Harrold, Jr., for
his dtrection and assis tance in the pre­
paration of this thesis.
45'7778
TABLE OF CCNTENTS
PAGE
CHAPTER
I.
II.
TII.
IN'IRODUC '!'ION
•
•
•
•
•
•
•
.
•
•
•
•
Gl!NERALI'J!ES CN 'tOPOLOGICAL GROUPOIDS
•
•
•
0
•
1
•
•
•
•
•
$
GROUPOIDS IRREDUCIBLY COONEXj'I'ED BE'!WEEN 'IWO
IDEMPOT!N 'IS
IV.
•
APPLICA TIOO S
.
•
.
. . . . . . .
•
Monothetic groupoids
BIBLIOGRAPHY
•
•
•
•
•
•
•
.
.
.
.
.
.
•
•
•
•
•
•
•
•
.
.
.
. . .
. .
•
. .
•
Topological groupoids with
QUasi groups
.
.
a
zero
. . .
.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
.
.
.
. . . .
.
18
33
33
38
43
48
"·
·CRAP'mt
I
IN 'IRODUC TION
groupoid
A
defined for
and a t the
se t
is a
every pair of
pertiee
many
Ore
1
to associativity.
[�]*,
[T]1
Frink (h.],
a
is
an
An element a
(1)
onl7
In
Chapter· 2,
etc.
•
we
a
group oid
if
(hencefo r th,
t opological
in
groupoids
first
this
their
for
pro-
motif,
that
s tr uctures ,
i.e.
Hausmann and
we wi l l
•
we will use the follov-
a
G
••• ,
,
is
use the
a
�n
•
common
,
e
0
and
has
r/ 0
all powers
gener aliz a ti ons
brackets refer to
I!!
V(all).
positive integers
or
a
are UDl-que.
will prove theorems r el ating to ideapotenta,
1'trUJ!lbers in square
end of this p aper .
proper­
of ·neighborhoods
for
property (L ,
a ,
•
abbreviati,on
V(�) [V(�)V(a)] n (V(�)v(ar) ]V(a)
e )
every triplet
·fr
all
V(a) of a
for
a
�n-1
said to haw
f} ])
Many of the se theorems are
at the
G,
topological
•
a
or
It
groupoid
of
�
r
m oreover, the
Our aim
[5], Etherington (2?],
is
a groupoid
of certain topological
�3 • ·�
a
a
of
V(�)
the
study topological
a ,
"'
• a
wi l l denote the empty s t
and
space, and,
We note, in relation to the
(0
n
to
element of_ a
iff introduced in_ [
,
G ia
If
ab
(22].
a • a , ·a�
t,' (L i:f' and
•
G is continuous in
�arri so n
"
i.ng. :not ation1
etc.
e G
dealt with non-associatiw algebraic
and Stein
If
b
to inv estigate the relati on
au thors have
Al�ert
a,
called a topological groupoid.
dissertation is two-fold:
( 2)
elements
groupoid
, �then G is
own sake J
which a single valued product
ti me a Hausdorff top ol ogical
same
multi.plic at ion .in the
��ace G
.
G in
of
theorems
numbers in
ide als,
in mob
the bibliogr aphy
2
theor:r.
wise
we establish the .following ·theoremss
For example,
bic o��pae t groupoid
tains at
G
__
a
.
and �-
2 • .3,
·tiona
gt'Ov:poicl
c<��pact
G
A is
_th en
is bicompac�,
G is
topological group if
groupoid, each llinimal left
pro�ded each element of
cloeed.
algebraic subgroup
bicompact o
'!'he
id eal and
ide al ,
G
If
(Defin!:­
is a bi­
which
is a bic011pa ct
each m1. nimal right ideal is cl�sed
K , the minimal
c losed
set or idempotents or
ideal
of' G
1
is
center
·
a t opolo gi cal groupoid
is
·
·�hapter
a
G is
ideal
K
denote
the
n
closes
bico��pact,
or
0
is
set of
� G so that
.
cormected..
groupoid
Ir
is
vi th a
a
G suc h �at
unit,
the minimal closed
groupoid vi th a
x
e.
i�.f'
I
unit,
groupoid such
each element of
_
I
is center assoeiat.i ve
cuts
I
·
•
I
in
o
If
G is
a llnit, no element
of'
eonditi on is satisfied.
groupoid vi th a zero, eTery left
has a left
G
a connected
(right} ideal of
a
I
let
exists a
there
is
a -certain neighborhood
G
G
or connected groupoids.
G
[17] groupoid with
..
disc ussio n
It
.
non-aimple
a
connected
xy • yx • u
then no e lement of
·-
with
e l� en ts . of'
is bicompact and
Tided
to pologi cal
topological
of a
.
... -
a
right ideal, and a mirrl:mal
2. 2}. Every algebraic subgroup
is contained in a maximal
associatiTe '£5].
a
a unique minimal c lo s ed
has
-
-·
I
is an algebraic subgroup ot
.
-rhese c oncepts ar e definea in Chap te r II
2.12, and
G
A
property � con­
element with
an
minimal closed
left ideal, a
subgroupoid.
closed
7
If'
Every bicompact groupoid
�ni1181 � losed
If
ha•
that
least one idempotento
��p��oglc�. gro�po�_d
gro up�
_
G
An element­
in
that
G ,
b i eompac t, connected,
I
It
G
cuts
G is
is
G
provided
a connected
conne c ted pro­
(right) unit.
In Chapter ni,
we
consider bico.pact gr oupo:l de
G
with
a zero
n
u it
and a
o�� �e
with
c
tf
�
,
0
cancellation law.
e
t/ 0
,
then
���rges to
terval
xe
<
n
�
.,
0
y,
which is
homeomorphism.
We note
and
e
G
€
y, c
x,
for
1
-<-.
x
E.
y , a nd
where
I
represen t s the positive inte gers ,
x
If
o
states that
an
t/ u
x
a,nd
n
that the unit
a
as
bers if the multiplication
from
f
i
G
.
or
has
'-n­
onto the unit
usual
the above theorem may be replaced
In particular, any continuous Dllllti-
the cancellation law,
obeys
of
the results
0
G.
of
acts
has property
[0, 1]
•associati�N-like" properties of
ties, we establish some
G
if every element of
space is the usual multiplication
acts �s a unit, �d every element
the
prop erv
has
is omorph ism as well as an order preserving
.
[0, 1]
plication on
G
ey
...
tud7
cy
<
center associative id empo tent .
next s .
- .
•
there exists a .f'tmc tion
multlpl1eat1on,
1
:x:,
ex
and which
of real number s under the usual topol o gy and the
[OJI 1]
by-- any-
!J
ar
u
and
ex
and
.r}
yc
1!1
If
y.
•
ye
xc
Our main result
tL , t� en
property
x
implies
{r't
�
� e�
0
which are irreducibly c�nnected between
r e al
num-
a zero,
as
0
.
:We
Usin g these proper-
theory for this type ot
of mob
gr-oupoid.
In Chapter
�oal
is
�
grou po d s
monothetie
IV,
with
we are concerned with
a zero, and . quui gt-o ups .
We consider the
a
a
t
is dense in
0
net (bn�n
�ere a� each idemp ot en t
is a c lose d ideal of
ply
AC
� (biti
eo_
·
A topological groupoid
•
iff there exists
finite "word s11 in
monothetic groupoids, topolo-
G
of
G
?-.
a unit and suppose that
E.,
G
•
.'!
P
such
that
We call
•
the c ol le ction
a
G
.
G
Let
U
Q
•
A
,
•
i
, and
AA .. • A
•
clue•
n}
�
is
A ·c G
0
gener a to r of
a
o
o
of all
� (setr definition --4ct3-) This n�t
·Ifbicompact, '"nC\.' {'bt,ai
such that
a}
0
1m-
be a monothetie groupoid with
where
P
and
Q
are
G
disjoiut,
closed, and non-emptyo
.
-
"words"
nite
-
in
�ny
-
.
and
a of
generator
-
a mono the tic
conditions for
P
Then
-
groupoid
-
Q
each contai,n infinitely many
G.
We g1. ve necessary and sufficien t
to be
a monothetic selld.-groupct
next consider the general theory ot topological groupoids
. If a
E, 0
pr �ded
. .
and
a8
��0
•
is
q
uni ue
A nil
lett
(right,
two-sided) ideal of
ments.
It
J.
G
or
for all n
(right,
then
a
called
is
two-aided ) ideal
G
or
We .
a zero •
nilpotent
a
a left
is
G which consists entirely of nilpotent ele­
G is elementwise bic011pact, every right (lett, two-aided) ideal
ia
either a nil ideal or contains non-zero idempotents provided
eT�ry �lement or
-
A
has property �
o
'!he groupoid
N-gr�poid it� its nilpotent elements form
� -�·� prop ert y A , then G
neighborhood
radical
1
with
G
ti-
(Def.
0
V0 or
4.9) of a
is an
an
is said
G
open se t.
to
be an
If eTery element
N-groupoid provided there exists a
which consisis entirely or :nilpotent elements.
bico 11pa ct
or
N - groupoid
1he
is open provided each nilpotent
ele�n� ·is right associative f 5 J. We show that any bicompact subset or G
_
is b�ded.
Finally, we consider the special groupoids vi th a zero men­
_
tioned in Chapter III . We discuss the concepts of nil ideal, radical• and
"arbitraril,-
We close
this
�eOJ.'T or
two
chapter
-
vi th
q�-gr-oupa�
that
ax • b
known
tl_lat a
•
.
�
eleaenta
�. xa
small• 'bicompact ideal neighborhoods
:ra
a
and
and
b
A.
an
exactly one
finite
implies
0
in
these groupoid&.
application or topological techniques to
gronpoid
ill
or
G
G
is called
a quasi-group
, there exists exactly one
:r E
groupoid that
G
such that ya •
sa�isfies
- x • y for all
a, x,
b
€
x
•
It
itt tor
€
G) is
arrr
G such
is
well
the cancellation law (ax •
7
the
a7
a quasi-grou.p. We
give conditions for a bieompact cancellation groupoid to be
a
quasi-group.
CHAPTER
II
O!NERAIJ:'l'IES ell TOPOLOGICAL GROUPOIDS
- - · J·- - -
_De
�Di �ion
. -- �·��i� on
ain e(i in
o ent.
2.2.
A
0
ot
is a subset of
0
for all
UX
• X •
XU
An
eleDtent e in
. e.
x
Clearly1
i.f
G
..
G
0
E
€
G
•
is a non-void set
of
G
ia a non-void set
(1:1) a two-sided· ideal
G
in
it is unique
G
an iden tity exists
in
0
X
an idempo te nt
itf
•
An element
all
is called
is termed a .!.!!! itt
Q
€
tor
G
(ro C T) J
!
u
Definition 2.6.
b
�
'1'
C
Definition 2, S. An e lemen t
• xO
.
elem en ts a,
which is both a lett ideal and a right ideal.
Clearl71
it a zero exists
.
.
iff
C
err
that
G
Detini tion 2.4.
Ox •
in which a sin gle 'Valu ed
(1) A� (rig!tt) ideal
eich
o
contained in
.or
G
aubgroupoid in a groupoid
_
such that TT
G
Definition 2.).
!
a set
ab is de fin ed tw eTery pair
procluc\
1'
A £OUpoid is
2.1.
E
0
and
is termed
is
an
an
ide��potent.
iden titz
ot
0
•
1
it is
unique and
is
an
idem-
potent�
�rinitioa 2. 7. A groupoid
b1
and c ··in
G
G
in which
(ab)c
•
a('bc) tor· all
a,
is. called. a semi-group.
Defini tioa 2�8.
If.
G
is
Jt
groU.poid
and
at the ,same time a Haus-
·
dortt topological space and) moreover 1 the multiplication. in the groupoid
G
Q
is continuous in
,
then
G
� th
var
iab�ee
w1 th
re�t to the topological space
is called a topological groupoi d .,
Definition 2 .. 9. A topological groupoid in which the multiplication is.
�..
associati'Ye.i� ��1� � topological semi-group
It
!1T 8
and
.
tor
bicoapact
are
bicoapact groupoid then
in
a
0
of a topological groupoid
subsets
T1 x
of
Cia
1
aG
1
0�
1'2
•
ie. bieompact
It thus follows that
!o and
Ga�
a
,
1
G
if
G
are bicowapact
•
We also obeerw that it
topological gr011poid
Q
'1'1
01,
gical gr�pold ��er
�·
of
and
Ta
are
connec ted
subsets of
a
T1Ta is ccnmected.
, then
A subgroapoid
The
�·
a
Br. Tyen onotf' e theorem T1_x T2
c onti nuous illage
the
-
is
-
a
!a
and
1'1Ta ie also lticOllpact.
then
ia
1'1
ar
a
topological groupoid
G
is itselt a tepolo-
r elati "fa topelo17.
f'olloving theorem gi"Yea a
s ta t elll ent
of' the asseciati "f8 1•
i11
\opolo gio al sroupoida in teras of' neigbborhoode.
.
..
�•or• 2.1. !. necessary � suffici ent condition �! �opol ogical
...
�·
groupoi d G � !?_!
.n
,!
�(a) [T(b)V(e_) � t!
-and �(c)
a
of
a
,
b
�
�
,
� !!! triplets
� c
Suppon the
Proof a
mente
1
, and
of
o
neighborhoods "T( al o:r
a
c o ndi ti on
G
,
gioal.
!?.! l'leighborhooda V(a) , _T(b) �'
respecti"Vell
is
such that
V(b)
[V(a)V(b) ly(�) n V (a) [V(b)V(c)]
On
{V(a)V(b) ]V(c)
topological semi-gr:oup !!. �
�.!!! a
satisfied ·
G :
there exist
ele-
'!hen one
•
can
tiDd
V(c) of' c such that
of' b , and
.' But this
• 0
and
(ab)o r/ a(bc)
!!
b_, c
,
the other h81'1Cl1 the condition i s clearlz
is a
ooatr �ictioa.
satisfied
a
tor
topolo-
aemi�gr-oup.
We will
Le ..
2.1.
ftbaeid.-voup!!
Proota
b
denote the topolocf.cal closure
of a eu s e t
S
� G !!_.!.topological groupoid �!!,!
G
·•
�
S11ppose
a
!!
!
E.
S
! oloaed subaemi-Eottp!!
and
b
€
S
o
Let
V(ab)
of
S
0
'be
G
!.!
'by !
!
•
an arbi-
•
7
trar)"
neighborho od o£
V(b)
o£ a
e.
1hen there exists neighborhoods
•
and b respectiTely such that V(a)V(b)
the:re exists
sa
ab
V(b)
Thus,
•
and sa
S
f,
81
Next, suppo se there exists
-
-1 a(be)
(ab)c
b
, and
-·
Y(c)
can
nten one
•
of
c
such that
e
e
, sa
T(a)
n
'T{a) [Y(b)V(c)]
all
a
,
b ,
-
-
and
, L� 2. �·
"'a2
•
.
in
.
[V(a)V(b) ]V(o)
,
•3 E. ·V(c
.
Y(a)
S
and
•
S such that
e.
V(a)
of a ,
n V(a) [V(b)T(c)]
of
ot
V(b)
• D
•
such that
S
� Thus a1sas3
e.
[V(a)V(b) ]V(c)
{ab)c • a(bc)
and
frYr
�
a
! topolopcal groupoid
•
;heii i
.!!!_!! � A !!. rig11t
is ! closed riiht
(left, two-sided)
•
Definition 2.10. We will make use of the f ollowing notationt
" "
" - a ,
"n
a • �
a
al
a -1 a
a2
a
•
t� haw
•
•
,
•
•••
2.11.
An
Property (L_ iff
element
a
"
tor all positive integers n
Examp�e � �1.
a
An
of
[V(;-n)v(�) )V(a)
eTe�t or neighborhoods V(aD)
not
�
"
€-
But,
•
ha-.a
ve
�
Definition
a
•1
and
V(ab)
!' ;
� G !!
(lett, _two-aided)
0
and
c
C
e
Hen�e, we ha'\'8 a contradiction
•
c
Analogoual7,
ideal of
Y(b) ,
.
and
neighborhoods
fi nd
s1
and ab
S
a , b ,
�t, there exi s t elements s1 , �·
11
such that
S
Y(ab) n
C.
e1aa
f,
and
V(a)
a
a
/\
topological.
-
,...
{1,
cr ete topolG£7 a
2, 1}
with
G
or
"
&r
and
is said
,
�� .
T(a)
ot
•
finite t opolo gi cal gr011pcnd which is
topological semi-group although eftrJ' eleaent has
b7 the set
groupoid
n V(�� [V(�)V{a) J � 0
o£ an , Y(aJ-)
and r
e:xaple of
•
propert7
the following Dntltiplic�tion table
CL ia liftn
and
the dis­
8
Leiau 2. 3.
G
·
po��ive
n
�d
.
�ntegera
_
2
X
2
1
) X
X
� Suppose this
r
r
and
11
__
may
x
�!!! .!!!!, powers
such
!!_ a
(a8ar)a
were
not
1,
2, or )
It !!!
eleMnt
be
�
�
aD(ara)
•
the
case.
" "
all
tor
1hen there
!!!·
positive
exists
J
"
(allar)a r/ a11(ara)
that
a
!!:!. lllligue.
A. A
It is first shown that
Proott
�n�'-��·
3
3
X
2
G !.! ! topological groupoid.
·Let
(2_�,
iiaa properiz
1
1 1
"
Thus, there
•
V(�) V(� , and V(a) or � , ;r �d a respec­
But this
ti��- such that ('V(�'V(�) JV(a). Q V(�n) [v(a!}v(a)] • 0
:
contradicts property � . �us, (aDal1a • aD(�a) tor all positi-ve inte-·
_
It is next shown that an+w. aD � for ·all posi t1
gars n and r
�s� neighborhoods
,
,
.
•
_
�
·
A A
A
·.
A
A
•
•
integ��s
n
and
m
tor .m
sume it is .. true
:• � �+l �(;ra)
- �
•
....
�s is true for
•
•
and
-r
(;-�)a
•
•
m •
pr ove �t is
�+ra
•
1
ve
•
by
definition.
true for
�+r+l �.
m •
'thus,
r
+
all
We
1
•
wi�l
as­
Now,
the powers
of
a
unique.
are
De�ni ti�n
i!_f for every
{a•an
nth
_
�
.
.
±}
power
,
of
a
2 . 1�.
� G
where
a
___
�P� 2�2:
An ex�le
is �ot bic0111pac't..
'
Let
X1' •
x
�
'•
1/2 tor all
{aDtn
e
I}
C
is t�e
r
�d
[a,i/2:]
l
•
of
a.;· are
an-
po�itive integers and
f!_f an
a
bicompact
ele•ntvise
(o, 1) with
G•
-
elementvise bicCIIlpact
is
G
�ch that 8.ll the powers
,
is �o�tained in
,
I
topological groupoid
A.
in
the
.G.·· Let
subset
of
unique,·
is
-�
the
set
�·unique
�:-
·
bi�OI!'P�t groupoid which ia
usual topoloa. -Define
'
a
6 G
,
�a7
·. a:·
<
lf2
• .
.
1hen
, .
'Jheorem 2. 2.
_
� elementvise
bicompact groupoid
-�!!.!!.
G
element !!!2, Fopertz a, eontaine �least �idempotent •
M
. Proof.
Suppo se
Lemma 2.3, �he po�rs of
A ":(
_
•
[an�_n � I} is
{air� � v} and
.
and
G
n�te �nterse�tion
is a positive integer.
n
U
{_A'(
•
Y"
t
semi -group .
{anan
z
�ho�se
Va(z)
of
7
C
n
and
z
an
�
}
I
,
�u e
Ti /
u �
•
V(Xa )
1
i 1
td
n
•
>
such
•
Since
Since
z
Ti
• v 1 - u ::::.. 1
•
1 1
y
Q
[anan
Q
•
is
and
(}
an
(
}
A to
•
, and
)
{!.tj
1
, D obeys
n
To prove this it
•
1here.fore, there exist
D
s uch that
Va(z)
or
and
z
such that
is bicompact
D
of the
n)
1, 2,
• ••
V(z)
be neighborhoods
. Va (y)
i•1
i
V(z)
. and
open set containing ·n
an
inte ger
e, D , there exist integers
•••
I
in
•
there is
,
}
f..
Suppose there exists
•
::> D
(1
V(y)
It will be shown
z
, and
V(y) C
E. D
(i • 1, 2,
and
x4
D
V(xa )
1
that
ld V(z41}
Let
•
-v(y)
Ti+1
or
Let
•
in
Xa.ljDV(xa)
Since
:J D
respectively
e D
x
Xa
Let
has the fi­
U
•
forma a group .
for every
y, V(xal
.o
V(y} Q • 0
tha t
�
of
n
[.lVCii (z)
V(z) (1
Put
{an
is a co��BU.tati ve cl ose d
1hen there is a
•
0
C
D
D
·
for all
D
•
-/
fl A
--<•1
Since
D
finite subcollection
a
so that
_
":�xa)
D
'7Xa
z
Va(z). n VaJy)V(xa)
XD
�
yD
�ueh t�at
neighborhoods
we. can
•
Since
•
�-
D
It remains to show that
, t!lat is,
�- _'fD
Then
is bicompact by virtue or
I}
C.,
G ,
•••
Hence it is cle ar that
is s ufficien t to show that
D
}
1, 2,
•
is a commutati-ve closed group.
D
in
'!hen , by
•
a topolo gic al semi-group by Lemma 2.1.
property and
the a ss oc ia tive law.
7
CL
As is eustolllal7 denote these
are unique.
the elemen trise bicompactness or
that
has propert y
a
represents the positive integers, is a topological semi-group.
I
Hence
a
an , etc., where
powers by
where
E
a
!!!.
Vi
avi €. V(z )
j • c , c +
l
,
•••
� 1
u
and
so
such that
for every
}
•
Then
Vi
•
�-
n·,cte> ,
c•1
element
n ACtc)
E
d
e•1
exists
a neighborhood
-
!Js.;tt;} ,
GD
d
�
"lhus,
...
"•
aTk
-
Then since Q is
0
V(d)
€
and
D
.
�
is a group.
D
.
LeBIIla 2�_k.
..
�
g
(2) �. some
then
y • g (.x)
__..__,
f.
......
€
��.t. U.
ag(a)
c
•
....
G
•
x
g(B)
x
vi th
:x
!!!:,
..
I!
.
I
_
!!_
r is a
Proof't
It will
ix • i for all
x
Then there exists
be
E,
z
A
Hen c e
, we obtain
•
is the idmtity or
This
•
xD
1
D
•
D
e2.
•
� � !!! powers 2f
a
!!:!!! !!1 A
.
�
{3)
A
(!
continuous function)�
g(x)
x
,
C A
B
C G '\c
• c
and sane
e
a
subset
compact.
If
�
xy
e 1
�, we have
�
x
!
be
� (1) i\XT !!
£
.::
H:�noe we may obtain �n _ open
•
U g(B) C G\c
If
[18].
a
B
n
U
XCI
in
I
�!E. algebraic subgroup_2!, �
compact
submob
�
by Lemma
A
�
then
,
I o
e,
I
2.,1. We will next
so that
Suppose there is a
such that
s,
Va(z)
z ,
Ya(z)
or
n
topolo�ical
is ! topological subgroup:·
sufficient to show that
one may fin d n eighborhoods
for all
!!!
is !. map
g
!'hen
•
Theorem 2.3 •
subgroup.
V(d) o
and we have � con�adictiono
. .
groupoid
e
!:. function
I.r not, then tor some
G
If
0
!E element
��
and
G
'!hen
•
t
a k E
such that
G � !. topological groupoid
A� G
t
\. g(B{.
includin �
_
Q , and since
��idempoten t element.
Let
c
Proot t
g(x)
"ttc
D , there
! necessary �sufficient condition �!!:.element-
.
� G
.!!:! unigue is
•
contained in
V(z) n V( y)Q •
� bi�ompact gcoupoid � �
-
Choose �
•
V(,}V(d) C V(y) Q
Corollarr 2 1
or G
d
V(z)
e
.x
or
C D
.
open set containing
an
!Jlere exists an integer
contradicts the fact that
tor all
r\ A(tc )
�1
and it is easily shown that
0
·
10
..
7Xe
xi
•
i
y S,
for all
i
Va(1) or
Va(y)V(:J:a)
•
Xa
in
y, aDd
a
S
:x
sueh that
for all
0
i
show that
i .
is a
A
yi
and
� I.
Thus
V(Xa) or
Since
I
xa
is com-
•
{vCxa;t):
paet, there exists a :rlnite s ubcolle etio n
k
i�V{s:Gi)
so that
v<xJ •s
the
may find neighborhoods
V(s)
that
n V(y)Q
such -t:Jlat
E;
a
V(z)
0
•
•
and
V(z)
we have a contradiction.
Ax
•
i for
It
where
is a
D
X
x
all
€
:V(y)
Now,
there exist elements
V(y)
f,
b
%A
'lhus,
.
of
i
topologl.cal
and
and if m > n
m €. D , then
m > m
•
(iii)
p e D , p
m
p
-,.. n
A
__
,...
and
is a function
�
v
if far any open set 0
d1
E.
such that
v
v{d•)
€.
0
d
e
a
and
A
V(y)Q
E.
i .
of
b
and
Similarly
the f\mc tion
A onto itself.
x � x-l
1hus,
I
set
;>)
(D ,
is a non-empty set
T
E'
if
then m
,
D
� p
n €
and
n , and
m,
D
(ii)
•
p
if
then for some
•
on a directed set
a E X
and
a , and any
D
If
•
·I
d E. D , there is a
QJ
v(d•) E.
d I E:
X
is a cluster point of the net
if given any open set
s uc h that if
to a set
D
and
(11)
dt
0
:::>
a €.
x
is a point
containing a
1
d , then
•
Definition
lett
D
m
if
containing
d1 > d
Of convergence or the �et
there is a
2.4 tha t
n � p
�s a topological space, we s�:r (1)
D
€.
a binary relation > satia.tying (i)
together w1 th
D
x
respectively so
b(b -la)
•
one
subgroup.
Definition 2.13, [9]. A directed
belong to
all
is a continuous function O:f
A
a
Now
•
is a group.
follows immediately :t'r<lll Le:nona
e
But,
o
i :for
•
and y
z
} of
k
. .•
i��V{xa.i)
-
Q
Let
•
2,
1,
•
k
and
'lhus
I
A
::J
i
11
2.1ho
(ri ght, two-aided)
sided) ideal and
A C:
A Dd.nimal
� (right, two-sided)
ideal such that if
L ,
then
A
•
L
A
ideal
L
is a
is any left (right, two­
•
We note that a minimal two-sided ideal or a groupoid
G
is unique.
1�
A
For let
and
AB C A
since
be
fl
B
2.4�
Theorem
�property
B
Q..
•
mtnimal
n
A
Thus,
•
0
two sided ideals or
an
is
B
{anln
g
Then,
e.
}
I
A
ideal and
� G �! topological groupoid
_
A
Then
•
B
•
£l T{n)
•
{antn
ot
8
r} .
ot
T
n.
is
that
D
Since
show
�
Y
ho�
w�-- may �hal!,
•
•
•
•
•
and
.
x
yx E D
�oup,
a
..
•
since
D
Thus
o:t
y
•
ba-1
{aDu� e, I}
gr:oupoid.
7J..
n
e.
x
2. 2,
{a8t �
0(ll:1')
D
D
2,
an
is
The r e also exists
•
• • •
o
xy
E
a
•
0
defined
v
2 .3.
�
a
is
group.
Next, we .
Let
arbitrary neighbor­
or x and O(y}
aY+iy E
an
E.
{ansn
I}
e
.
But, there exists
•
Thus
.
two-sided ideal of
a
that
find neighborhoods O(x)
Y • 1,
is
Suppose
•
B
€
D
6
•
}
I
O(x)
O(y}
or
for
Hence
•
Similarly
•
Since
D
is a
�� cannot contain pr operly any ·two-sided i,dealo For suppose A is
Theorem
Let
_
n
or all cluster points
is the set
topological group by 1heorem
C O(xy}
for
two-sided ideal or
But ,
a
E
one can
'!hen
&"+iy+n e, O(:xr)
�
...
t�-sided ideal or the mob
· is ·a
_rr...
1, 2,
{l, {•1•1 Y}
•
as in the proof or 'lheorem
y .;.s��h that O(�)O(y)
Y
_
A
group � !_! � minimal � sided ideal
tJO .
C?mpaet, it is
{ant!l f. I}
or
D
'Ihe set
Proof.
-/ Q
!!!, a ar e
(��the powers
!! compact, � � � all cluster points � � .!!,!
f�rme ! topological
B
and suppose
unique)
an
•
n
o
y
Let
E
D
and
e:
A.
.
By the remark aboTe, such.
2.!).
Hence
A
•
D
Every compact
•
Thus
groupoid
a
€.
A o
Then
ya
•
b
·
E
·tet C2 be the
{Aa1 e."}
a
is a minimal two-sided ideal
an
ideal is uniqueo
0
hae !. minimal closed
family of all closed subgroupoids of
be any nest [9] in
�
A
D
!!2_­
_
P.roo.t'.
•
D
o
Since
0
·•
"'/... has the finite
•
13
intersection property,
Q A4
mtnimal
C Aa
for all
pri nci ple
0
a
e 1\
and hence is a clo sed groupoid.
�
'!bus,
•
r 9].
2.6.
'.lb.eorem
n Aa r/ -
a: 2../\
has a minimal member by the
G
Every com pact groupoid
contains the unique minimal
closed ideal.
----
Proofs
?"l_
{BY"'
•
:
e, 1\
Y
}
is a closed ideal of
'l'hus,
2. 7.
Theorem
ideal and
!
.
in
i s contained
Prooft
0
�
x
�a
•
B
c
Let.
Aa
A�
Let
•
us
A
be any
ot'
x
•
Aa
�
B
1'
Aa ,
x{yz)
·
_.
nten
A�
S
xy
y
Aa
for all
•
Y
for every
G
� 1\
•
preT.i.ously.
�.!minimal closed lett
! top ological group oid
G
algebraic subgroup of the topological
-
E.
C
Let
a
-·
C2 be the family of all subgroups of
consider
1hus
•
We may assume
(:xy) z
0 and hence
!. maximal algebraic subgroup.
Let
and
E.
Then x
Q BY';
uniqueness was established
alS!braie subgroup or
partially ordered by inclusion.
{2;
By
C
Every compact groupo�
�eorem 2.8.
�oupoid
Q BY"
But,
•
1hen,
o
Let
The proof of this theorem is similar to th at given for
2.6.
-
G
Q,
•
minimal closed right ideal •
. .
Prooft
1heorem
� nest i n
be
minimal_ member_. ��
a has a
G
(2 be the fami l:r of all closed ideals of
Let
But,
e
B
•
(j
A�
A�
A� C
e
c
and
,
__
.
{Aai
•
{Aata e 1\}
tor s ome
e
'7
Let
A-..r
a
B
€.
z
•
Supp ose
•
A-..r
be the identity of
1\ and hence
e
for some
a
x f: Au
,
y
y, . and
A
•
a:
z
Then
is an identity
t,
I\
B
e.,
.
z
are in
� , and
,
e
e
B
•
1\
and
i s the identity
or B
Therefore
v
�n A.,
are
Q,
Then
•
We may assume
•
A
l;>e any chain in
X , y ' and
for some
x,
Hence
x
/\
<c.,
�
/\}
e
Let
.
and
a:
Cfntaining
G
•
Let
x-1 e Aa
C B
•
�d xx-1
f�� al�
e
=
_
� I\
a.
Now!.. suppose
memb$r
x-.1%
=
0
A
o�
1hen, B• ::> A and B'
•
t:Z
•
C
.::> !4
a maximal
has
algebraic subgroup o f G containing the
an
is
[9),
lennna
H�_nce, by Zorn's
•
B'
B is a group containing A and B
Thus,
•
member.
maximal
•
Theorem 2.9. !! G !!_!. compact topological groupoid, eaoh !.!!!­
!!:! algebraic subgroup � G !!, .! compact topological group.
Proofa If A is an algebraic sub group, I is compact and hence
_
is a topological group
-
-
D$f1ni tion
•
.M
Theor•
in
G
if
-
x(cy)
an
-
c
• f
x 1
••
y
€.
G
•
G
of a groupoid
for all
-
A
is maximal,
element
(xc)y
a
...
A
Since
2 • .3.
-
We shall call
2.1S.
center associative
..
by
_
� [ 5)
borem 2 .10. If G_ is !. bipompact groupoid, � minimal !!!.!_
·-
,.
�
'
I
�
'
._ideal and each minimal right ideal � G !! closed provided � element
.
of K !!!!_ minimal closed ideal � 0 , !! center associative '!.!! 0
.
-
.
,
•
G
Prooft Let L be a minimal left ideal of
-
-
a
E.
C
Ka
•
L
for
L since L is a lett ideal. b (Ka) C (bK) a C Ka since every
element
or
and Ka
•
-
Si��S.:ly,
K is center associative in
G
Th.us, Ka is a le:rt ideal
•
L because of the minimality of L
. .
·- . -
.
..
i.t
R
is a
a right ideal contained in
R
for a €
we show that
minimal
Hence, L is closed.
•
G
right ideal of
R
•
aK
Hence
•
R
,
aX
is
and R is
closed.
'l'heorem 2.11. !!!! � 2.! idempotent&
E
!!
.!
topological gronpoi'!
is closed.
----
Proofa
p � i
.
rvs_p) J�_n
It
If' !
•
0
1
the
result is trivial.
Suppose E .f 0 and
p ; p2 , there exists a neighborhood V(p)
v(p)
··-
o
. stnce
idempotent f exists in V(p)
P
•
c,
i
, -v(p) _n
_
!his contradicts
E
;
o.r
0
p *uch that
, i.e.,
rvCp) 12 . n
v{p)
an
.;
a·;
lS
p2 • p ,
'lhus,
e
p
E_ , and
We next discus s
topol ogic al groupcids
E
is
c losed.
This will
�
be
co nne c tivi ty as related
or
briefi:r the concept
discussed more :f'ully in
Chapter
�· � � minimal closed ideal K � G is connected.
1hen, C • uC C GC
Proof: Let C be a componen t or K
•
is a connected
·ac
is c lo se d,
K
Def�nit�on 2.1 �·
I
denote the set or
"1'
e, G
A and B
If
G is
Let
�
x
that
I
be
to
due
i
...
!. conne c ted
� which�
Theorem 2�13�
pact
and_
!.!£!!_ element �
-
-·
Prooft
�
sume
element
{
__
[18•
s
is
and
of K
0 •
•
•
Theorem
A n
G
in
Q n v.;
�
�
po nt s.
B
Q
cuts
and
I
!_
if
Cl
•
1] o
2 let C 2!_
-�!!; �
C
!
!.!. center associative �
CO!ft­
G , ��element
•
• u
,
center
g(s) •
associati-ve.
x·la
for
eiatiw property, it is easily seen
Let
X
s
€,
all
that
r
e I
G
is a
•
we
'Ihus
the result follows immediatel7.
·
or I
•
P.
Hausdorf:f' space
� X
C
i n
means
non:degenerate. W� r��st note t�at I is
is
:f' s) � x_
If I
Ddnimality
G !!, !. connected groupoid � � I !!.
!!.
o :f' I cut s I in G
Sine•
I iff there exists a
A\B
Wallace
_ _
ne the r
·=c.·�
�oapoid with_� unit, let
Let. us say that
! compact �ubset � I with !! � �
�points,
�
, Qn u.;o
utv
2.4.
topological
a
C
Simi.lar;J.y ··CG "
•
K fro m the
agree that
us
We next state a theorem
Lemma
GC • c
C G!C
•
are separat ed.
G-P•JJUV,
C •
thus,
xy • yx • u
so �hat
K -!.-
or
elements such
Definition 2.18.
i.e.,
subset
is cl osed o
C
II:r •
!!!.!, G � !. compact, connected voupoid !!_!!! !
1heorem 2.12.
Since
to
a as-
m y
a �oup since e�eh
and
1Jsing
X
,
the
u
�
·
n��ne
center ·�-�o­
homeomorphism o:f'
··
G onto
K
.•
16
itselt.
I
Since
ciative ,
• I
f(I)
groupoid and each element of
a
is
Now assume
•
f(u) • x , each poi nt ot I
2.1J.
leliiD&
0
1
Hence,
.u
g(x) • u
then
Definition
contains no
u
cuts
I
in
cuts
does not cut
cuts
I
0
in
I
in
2�14.
It
"'V(y)
tripiei ,2! neighborhoods
7�'
reepectivelz!!?!.!!!
�
I
K
We note that
B
are each
then since
B
B
�-
� • u
reasoning
C
xK
u
�a�
__
as
t ,
is non
K
[6].
.
1
€
xA.
x
g 1 _G
€
1hus,
G ,
G
,
r
x •
s •
by
tha t
�
G
t�9(x)
has
cuts
1
mple ,
G
•
G
r/
may assume
has
1
•
•
in
G
in
•
·itt '"'G
[17]
�simple
hence
and
B 'u
gr:oupoid
xA
r/ 0
E:
u
G\.K
cuts
1
and
A
and
•
It
:x:
€
.
has a point in both
e
yt
a right inTe ns
.•
u
If
K C A\ u.
that
,
A
G
1
B ,
and
i.e.,
Utilising the h:ypothesis 1
a lett inverse.
and applying elaeRtary, algebraic techniques
sees that
s)
I
to be simple
K
A '-.•. r/ 0
t{s) • x
where
I
.
9(x)
has a uniqu inverse
x
cuts
I
virttte of 'lheorem 2.12.
x
and
•
s
r(t(s))
•
G
1herefore,
s
onto 1 tself.
eaeh
•
Let
9{z)
g(s ) •
and
Clearly the maps
i� a ���om.orphie� of
r(u)
si
u
•
We
1� Theoram 2.1,
ziDg the hypothesis , og.e
e
e,
cuts
� V(s) !!!, y , x , and s
G � � yx • u, � �
,
the connected set
Siud.l�l:r,
easily
o� sees
G
continua·
lherefore,
•
x
f., I
n [V(y)V(:x:)]V{s)' f .q !.2!:_ every
s €
and
o
A n B
,
V(:x)
,
x ,
is connected
U
A
G •
G
· Since
Proofs
then
x
If
•
G is said
�!�such�_ Y(y) [V(:x:�V(s)]
element or
G
!!, !. compact, connected,
G
asso­
center
1hen si�e
•
proper closed ideal.s.
'lheOra
is
But, this co ntradicts
•
Thus , no
•
A groupoid
2.19.
G
0
in
I
I
•
s
and
g
B
utili­
and
it
are c ont inuous.
Hence, since
point of
•
g(t(s}) • s
and
f
� consider the
u cuts
is a cut p�int or
'
17
G
But
•
has at least two non cut points of itself.
B
Then it is easily shown that
one of them.
contradiction and hence
g {x)
then
G
cuts
• u
Ex�le 2.3.
u
Hence
•
no
x
•
Now,
if
I
cuts
in
G
cannot cut
e
X
G
•
·
E
B' u
This is a
G
cuts
I
.
.. Define
- U8llal--.,; topology.
(O,. l ] aob • ·a2o where the
.
asserted.
as
multiplication on the right is the. usual multi-
.
of' re�l �mbers.
[0,1] with this nmltiplication is clearl,- a·
topolog;!.cal. groupoid but not
a
It is non-simple,
topological semi-group..
connected, satisfies the nei ghborhood condition of the above theorem,
I
cute
·
and
I
I
the following multiplication on
a
is a left unit,
be
[0,1] be the mit interval of real numbers with
Let
the
p!f cation
G
cannot cut
p
p
Let
(with respect to, 1)
•
1
1
Obviously, no element of
•
[ 0,1].
2.1$. Let G be
Theorem
! co�nected groupoid
� !.
(1) If G � !. � �� eve:ey left ideal �
(2)
If'
G
!!!:.2.•
!! -connectedo
G
!!!:! ! right unit, every right ideal 2!_ G is
�-
neeted.
{3)
!! G has !. right unit
_
is connected.
-
(1) Let
Proota
x
E
Hence
x
E.
Hence
Ox
L
<;
L
.
�
Hence,
R
�
R
Hence
R
•
'Y.LGx
x
·
and
� unit, every ideal � G
0
E.
be a right ideal o f'
•
YR:xa
X
G
It
•
x
Gx _for all
€
L ,
x
€..
x
€.
R ,
tor all
x
€
L
•
is connected.
Ox
is connected since each
Remark.
2!.!
be a lett ideal of'
L •
(2) Let
and
0
e.
G
xG
If
•
�
R
•
'. '
xG is connected.
(1) . and·. (2)
.. We_ note that 0 could have been replaced by lett zero in .(1)
(3)
__
L
is connected since each
xG C
R
.
The proof of
and by right zero in (2).
{j).
is
'similar
to
those g:t ven in
•
CHAPTER
CROUPOIDS IRREDUCIBLY
p��nte
Definition 3.1.
�. and
b
IIT
CCNNEC TED BE'IWEJ!N
A space
S
b
it and only it it is connected and is the only con­
and
b
•
a
and
is a cut point, separating the space into exactly two components, one
and the other containing
order relation in .S
b
S -
in
e
every x
.
y
by defining
<
S
_ _
}
x
X
i.f
We.. can introduce a linear
.
t. c , c the component
"1
x
E
S , the set A(x)
is pr�cisely the component of
-
b
�
S , by-replacing
-
x and the set
in
S
- x
vi �h
e.
�
a
S
,
B (x) '•
.
..[21].
...
tor
y
}
:>
x
{ycy
€
s,
It is clear
such that a
S:
x � b
a tor b in the order relation.
·-
ibis order relation induces
top�logy,
x
{7ay
•
that we can define a reverse order relation in S
tor every x
S
b
•
�� pr e cise l ! the component or a in S
£.
b [21] o
x [21]. Under this order relation,
Moreover, .for any
y
a
such a space is Hausdorff every point different frart
containing a
or
two
is irreducibly connected between
nected subset of itself containing both
It
'IWO IDEMPOTENTS
a
topology in
basis elements _or the form W
•
S
{y:s
, the so-called order
<
y<
t and
s, t
E
s}
,[21]. Since we wish to use the order top�logy in this section, w� prove
that i.f
S
topology or
_.
is compact, the order topology is equivalent to the original
s
Consider
or
s
Since
,
•
ta
(S, U)
__...
(S,
y) where U is the original topology
. ..., is the order topology,
Y
C t1
[2ll · and since
S
and
f
is the identity function.
is compact,
f
r
is a homeomorphism ex-
tabliahing our assertion that th e topologies are equivalent.
19
Now,
S
and
S
suppose
the
two
relations
order
We shall
It
S
. ponent �
c
S
-
; 0 r/
x
, x
if
s
and
c
It
�
S
C) ,
-
s
<
- y
•
<=
<.
A
onnec ted
�s,
t]
__
is
poin ts
Let
the
S
� y
s
so
that
!:.
t
}
�
0
x
�
u
•
'lhe set
•
b
t
<
a
y <. t
�
a ,
<..
s
U B
whe re
A
€.
X
T
¢_
T
}t.
T
T
B
A u B
- X -
€ T
x.
, then
•
T
�
1
[s,
t]
be the two non cut
and
y
B
p
C
and
T hus ,
we
and
[s,t] such t hat
€.
are
B
To
oin ts of
y
is
By
·Y
t/
•
'!hen,
a cut point of S •
Hence
separated.
•
S
•
A
Since
A nor B contains both s
_
not irreducibly connected between s and
A
S
is a connected set contained in
_[6], neit� er
C T
s
if
Further,
•
�
A
which implies
both
S � that
is ! connected subset of
n
Let
•
� C a com-
! compact Hausdorff space irreducibly .2.2!!­
If
and
a
connected set meeting
[s, t]
.
T
t ��points in
Proof.
have b
S ,
E
W e shall choose
•
.! connected Hausdorff space
� S be
nected between �
S
€.
n (S
T
Lemma 3.2.
b
S be
Let
!!!2. B !!! separated, and g
A
"1'
u
in
defined
have
{
•
and
a unit
zero and
is compact and is irreducibly connected between
follows easily that
Lemma 3.1.
n
o-
with a
[21].
t
and
we
[a, t]
denote
J s, t) is co mpact if
�
topological groupoid
is irreducibly cormeoted between
one of
s
is a
L emma .3.1,
s,· t
and
t
t
Thus,
•
Y
.
B
and
e.
ar
e
Otherwise,
•
T
T
•
is a
Hence
•
Lemma 3• .3.
Let
S
be ! compact Hausdorff topological
is irreducibly connected between ,.!!!2. points
be subsets of 5
uch
s
that
a
and
b •
pa c e !�
s
Let A
and
B
I
20
(2') A n B
•
0
(.3) A ; 0
I
B ;
{4 )
..!:!!!!. there
a
�
A
y
and
__
�..,. �
for all
--s
Since
��ppose -- � n B r/
0
exi sts
B
ment U ( '(' )
�
€.
A n B
a
that
">
such that
a
"<
Y3
€
V'
B
u{-y-) n B
"'Y"3
<:
•
n B ;
Y
a
•
£.,
e
a
0
Y1
and
for all
A
•
Y E.
..
A whereas
Corollary
-ti,
3.1.
11!'-.
Y'a
•
•
Supp o se
Then there is a basis ele­
0 . This
•
n A
< "Y
for all
such that
E,.
�
�3
•
for all
B
contra­
�
a
ftr all
et
Suppose there exi sts
> Y
�
Yi. .::: J)
sue� that
�
A n B; 0
or
A
and
B
<:.
•
a
�
� B
� Y"1
1
containing Y
•
V'2
Vi C:::: Y12
•
Y3
implies
Thus
V)
E.
B
•
B
•
Hence
Suppose there
Suppose
"2
A such
E
y e.
contradicting the fac t that
A and
<
as sume A n ! r/ 0 and let
Next
0
a
.S. Y
a
Then
•
U(Y)
He nc e ,
Y e.
�
•
A
such that
i .
V €
� � Y"1 -! . ys
tJxiste
�
such that
1
such �
I nB
"< e
B
E,
�
ntus there exi sts a basis element U( v)
•
are two elements
_ _
Let
Hence ,
•
� Y for all
and.
.
� e;
� � V tor all
B
E.
�
is connec ted,
containing
dicta the fac t that
Y €.
!!!2.
£ A
a
if
0
is � {and only one) Y E. S
Proof.
there
A U B
S •
{1)
a
for all
A
Then there
•
< Y'2
E.
implies
Thi s contradicts 2).
Under � hypothesis 2!_ Lemma 3 . 3, either A eon­
taine ! large st element � B contains a smalle st element .
'Proof ...
Y
It
Y'
� A , Y is the largest element in A
is the sma�lest _!lement in B.
By (1)
- -
_
If
y·. E
one of t hes e cases mus t occur,
'
while (2)
•
implies both c annot occur toge ther .
B
21
Let S
Definition 3 . 2 :
two
� onnected be ?'e�n
a
be a compac t Hausdorff space �rreducibly
points .
Let
E
be a subset or S
E.
y --� �-- such that x � y .for all x
.
7
bounded below, . and call
Definition 3 . 3 .
.
a
Let
E
E
lower bound or
T.
(b)
•
If
is o�
>
x
is
•
Let
be a subset of
E
Suppose y has the following properties
E
E
, we say that
b e a c ompac t Hausdorff space irreducibly
S
connected be � en two points .
of
If there is
•
(a)
S
•
y is a lower bound
E
y , then x is not a lower bound of
the gre atest lower
bound
E
of
Then
•
[clearly from (b) , there is
a� �o �t one such · y . ] We shall use the abbrevi ation •int. • or "g .l.b . "
�
. .t:� "gre ate� t lower bound"•
LeDml&
_3.4. Let S be !' compact Hausdorff topological space �
is irreducibly connected between � points
void subset
may
•
S
-
B
E
take
i�
it
auf' .rices to
A
and
a
iff there is an
of
lower bound
prove that
A
E
x
�
e.
B
,
a
s
x
•
E
is a lower ·bound of
•
Let
, and every
To prove the existence of the inf . ,
•
E
Since
We now verify that
� E
there is
'lhus,
'lb.us
•
an
a
<.
y
e.
A
x E.
E
such that
p
for all
, and
a
�
A
B
is non-de generate,
is bounded below, there exists a y
tor all
�-· > x
such that
sati sfy the hypothesis of Lemma 3 • .3 .
B
E
E
�
x
has a greatest member .
Evidently (1) and (2) hold .
Since
Let B be the following set
t o be non-degenerate .
Clearly, no member of B
A
Then !!!!:l �­
c6nsists of a single element the result is immediate .
C. B
�
•
�mber of
E
If
or elements :
A
•
or S !:!! _.! greatest lower bOllD.d .
E
Proof.
'lhu�� _w�
a � b
e.
S such that y
r/
�
0
x
r/ 0 . Hence (3) holds . If
� :;:> x
A
and
•
If
� � B
a
e,
A ,
and (h) is
•
22
satisfied.
Hence, by Corollary 3 . 1, eith er
contains a smallest element .
Le�
p e., B •
_
>
�'
•
X
Definition 3.4.
cancellation law iff
y
G
�
o
,
·with
c
-/
0
such that
>
�I
X
�
>
�' e
,
G vi th a zero is
A groupoid
:xc •
x
Choose
o
s o that
B
ye
or ex
said
cy implies that x
•
to satisfy the
,.
7
tar
all x 1
•
!. c ompac t groupoid � !. !!!:2.
.
s atisfies .:!:!!.! c anc ellati on !!!•
<
ex
A
\.
•
.
'!hen
x
<. y implies
cy for _all x , y , c € G � c r/ 0
_
Proof . Let :x: , y £ G such that x <. y .,
�
!.
�
0
J,et k.
�
inf' ,
•
pose k
:>
.
0
yc
{xa \
•
C,
for all e
}
C. A
Xa
[xa1u ]
}
Clearly,
•
We next show that xk
l.� espeetively such that
There is a n eighb orh oo d
C V{yk)
•
a
E.
�
ne i ghborhoods V(xk)
V(xk)
hood 'V(k)
'f
and
•
k
Then A r/ 0
and
of
k
Thus xk
V(yk)
<.
Suppose xk
'f €..
V'(k)
yk
such that Xf(k)
V(yk)
>
a
A
•
a
XXa
>
and
Then
yk
> Y
.•
and
l'Xa and
y by the cancellation law
Hence there exist diaj oint
•
<. V
C V(xk)
e:.
yk •
xk
of
imply
Hence ,
o
x •
or xk and
imply
u
exi sts by Le111111& 3 . 4 . We sup­
yk •
yk , then
•
since
such that xV· (k) C V (xk)
Xa: e
an
If xk
and V(yk)
€
and
of k
V(k)
and we have a contradiction.
e
V(xk)
But there is
we have a c ontradic ti on .
a
yc
Le t
there _ exist di s j oint open neighborhoods V(xk) and V(yk)
yV: {k)
<
xo
•
{xatxe
-
�•
� is not
� � 0 i s irreducibly c onnected between 0 � u • Suppose
.
B
or
This completes the proof.
� G be
'lheorem 3 .1.
� E
x
Since
the smallest member of B •
cont ains a largest element
We prove the sec ond alternative cannot hold .
'!hen there is
>
p
such that
A
respe c t� vely such that
yk
•
But, there is a neighbor­
and yT-(k)
C
T()'k) • We
23
may take
V (k)
Xa
e
V(k)
,
c
e,
V(k)
•
xc
<.
ot
k
an
Xa
for
yc
< ye
XC
for all
[0,
t
r/ 0
for · all
- There exi s ts a
Henc e ,
•
c
to be an open int erval about
k
c)
e.
c
0
•
and
Let
xc
.3 . 5. �
G
satisfies the c ancellation
<.
t
€
G
•
A
yc
t .<:
such that
A.
e.
<:.
XC
k
ye
for all
Henee·
•
This c ontradic ts the definiti on
and suppose
-/
c
0
'!hen there is
•
simi.lar argument yields
� cy
ex
be
!
groupoid �
!!!1 G
�
.! �
!.
�·
0 and
� idempoten ts except
c ontains
G
If
•
Proo.:f' .
t
c
.
But,
o
an
•
Lemma
u
an d
[ t, u-)
•
v · ( k)
€-
t
Since there is
o
(xa, U )
£,
C
k
•
.:f'f
• _,
..
Let
.:f'u !. and
.
hence
Corrollary
which !!
be an idempotent of
f
!f.
3.2.
irreducibly
t
u
•
.G
G
t r/
and suppose
0
'lhen
•
...by t�e cancellation law .
!
is
!. �
compact groupoid with
0
connec ted between
and
�!�
and whic h satisfies the
u
c ancellation law, then
(1)
ry
<
(2)
it
x
.s
(.3)
i:t
X
� property (L
min (x,
y
_
idempotent .
to
Proof.
xy
� x
by
yw
implies
pr operty
{
:z:ll t n
Lemma
x
Theorem
�
� ,
�
.3 .4
·
(1) •
r}
yv
u
y)
�
!!
w
X
:::. v , then
-j
U ,
{
n
x sn
1
.!d:!!!!.
xw
2.,
{7fl&
·
I}
yv:
s.
.!2_ !_!!
c onverges
n
e.,
y
s.
t}
c onverge s
0 .
� u
3 .l o
(2)
by The orem
� x -;;,.
xf
implies
x
3 . 1.
�
xy S.
!: y
impli es
Henc e
r'J
. . •
and
y
xw
xw
5
yv
by 1he orem
must clus ter at an idempotent
t
that the monotone decreasing sequence
•
.::::
u
yw
(3)
•
3.1.
impli es
and
Since
:n
e
r}
v
has
x
By �e orem
But, it follows
�
S..
w
2 . 4,
from
has a lind t .
.�
7fl •
_Ir x r/
{
,
I}
then . . :z:ng n 5
is bounded away from
Thus by Lemna 3o 5, lim x!l • 0
and lim xD r/ u
n
n
-� 3 . 6 o � G b e .! compa� groupoid wi� ! � � !. �
lim
Henc e
u ,
n
t
o
u
o
o
which !!.. irreducibly c onne c ted between
cancellation !,!!
and
E
O
.
o
·
and
3 . 7• .
Lemma
Gx
'Ihus
•
•
[0, x ]
� which satisfies .!!:!!
•
s
€
He nc e ,
•
n
time � .
_
each
ous ,
.
E
fn ( l:l'tJ.) C_ G
y into G
G by
6n --+-
fn s
define
(We
•
by
or Lemma 2. 3)
vir tue
c an cellati on law.
.
Let
rem 3.1 �
i f x r/ 0 ,
assert that
.3 . 1.
1heorem
:x, y
-
•
Suppose
•
Henc e ab •
aa
a
0
•
It follows 7.'!1
Suppose
<
x
S.
nten
a, .
x
has
1
0
-
0
and
fn ( 6�
,
E
G
h
•
.;
G
an
'!hen
o
a
�
u
a
y <. x
unique
o
The n
x "P for all positi ve
�th
2
y
x
where
are
G
Since
o
This i mpli e s
nt h root.
�..: ab·
s i nce a r/ 0 ,
1
is continu­
fn
S ince
o
root.
has a unique square
b
'lhus,
•
By induction,
E
satisfies
that all powers of x
note
is a cormected set c ontaining
x r/ 0
G
G by fn (y) • xD
__
•
• [0, x]
semi -gr:oup.
fn is onto �d� in particular, that any x
pose a2 • b 2
Suppos e
o
G !!,!! property
is irred.uibly connected between 0 and u
We now
xG
be the diagonal or the cartesian product of G
n
is the proje c ti on of
unique for all x
and
[0, x ]
e t
!!! abeli an topological
�n
Let
For
0
� G � !. c o!pac t groupoid � ! � and !. �
G .. !!.
Proof .
c
z
By' a similar ar gum n ,
•
� cancellation !!!, � every element 2f
�
o
• ax for some
Z'
Gx 1
which !.! irreducibl.z 4&0nn.e cted between 0 � u
is � �
G
·
Gx t� a c onnected se t containing
Since
X by C oroll ary 3 . 2 .
=
Z
u
xG !2!_ all x ·in
•
�
C: [0, x]
Gx
0
e
( 01 X] .
•
0 �
Gx ·:J [01 x] by Lemma 3 . 2o If
,
x
Gx
Then
• .
Let · x
Proof .
a
24
b2
�
a • b
by
Sup-
'
•
a
2>
the
root .
<
x '�
integers
by' Theom
o
We
c all
this result (1) •
Let us define f'or x
€
G
, x r1
0
p and q are positive integers . It is inmedi ate from the fact
that the ..t�th roots are tmique, tha t r is well d efined for any positive dyadic rational r , regardless of the representation of r
By
the same
gument , straight forward verification establishes th a t we
where
�·
x
•
ar
e an
{ 2)
We claim further that i.f r and s
� , then r �
•
'lhe�e exists a
rationals such that
dyadi c rational t such that
r <
rt xr xt By Corollary 3 . 2,
r t • s By (2) ,
xrxt � xr . Therefore, if {rn} is' a monotone_ increasing sequence
or dyadic rationals, {rn} is a monotone decreasing sequence in G
For
E. G
x r/ O, u let D �r :r is a p�sitive dyadic
r ational} • By (2} , D is an abelian submob of G
We claim D G
Let V (O) be. any neighborhood or zero. Since r , xn� 0
b7 ���llary 3 . 2 . 'rhus , V(O) n D rf 0
Let V (u) be any basis ele­
ment containing u
Let y e V(u) with y < u Clearly 7n� 0
�.D
1}hus there exists a positive integer n such that r < x He nc , by
{1) and the linearity of the ordering y < x1/fl and V(u) n D rf 0
Le t t be an arbitrary element of G , t rf 0 and t rf u and. let
B
{s : a -� s <. b} be an �bitr� basis ele�nt in the der topology
containin g t
W'i thout any loss of gen erali ty
may assume
are dyadic
8
x
+
..
8
x
•
•
•
•
•
8
•
x
•
X
,
�
m
•
•
X
•
u
.
•
•
e
•
•
•
I
or
•
•
,
we
•
•
a
0 <
< b < u
R'
'
l . u.b .
a
maeta
r
a positive dyadic ration
are non-empty o
Since
B
r' '
and
sequence as
r
r'
incre ase s ,
�erefore1 we as sume
•
a
is
n,
By .Le111111a
to
c
.
.
3 . 4,
; Let
w
�
u
E
or
r'
rv '
•
.f+P)c �
for s ome
r
Le t
o
{Pn}
e, V
•
But, ·
•
There exists a
+
Pk ::::>-
�+Pk
a mob by Lemma 2 . 1.
•
i s a mono­
convergin g to
r'
{?n} ;?! b
n
Henc e {:l }
countable dense set and since
G
•
w
s·o that
C W
Pk
r1
�
r•
.f
•:
.z!!k
1
•
Since
C learly
G
<
r'
•
,
be a monotone
n
� onverges
[0, a ]
and
n
D
'!hen
•
for all
U
U
•
0
•
V
r/
0
,
{Pn} such
in the sequence
This implies
•
e
uv
C w
w
and
From thi s c ontradiction it follows that
__
tween two points,
b
c
UV
such that
�'!s � . i�
•
?
c
be the
r'
If'
•
-
[0, a]
numbers
S. r ' '
r'
{r}
Si nc e
•
R'
open set c ontaining
V and r
[o, � ]
has a g l,b ,
�
joint .from
_
Let·
x8,
r �
implies
multiplic ation, there exi st open s ets
e
U
z'Pt e.
that
an
be
and
C.,
U
{�
_
-
w1 th
al}
monotone decreasin g se�uence such that
By the continuity
. ,
<. s
r
R"
be the g.l . b . of
increasin g sequen�e of dy�die rationals of
{zl'n}
rational}
a positive dyadic
have upper and lower bounds respec tively.
R'
of
1
R'
and
decreasing
tone
D
•
R' '
and
r
.
R'
Clearly,
R'
{r_s_r � b ,
{r t� � ,
•
R'
Let
•
G
is abeli an .
is dis­
:iJ • G
Since
D
•
.
is a
is c ompac t and irreducibly connee.ted be-
is hOl'lle omorphic
to
I
1
the unit
interval
of real
[ 21 ] .
hample
3. 1 .
An example o r a finite groupoid wi th
a
zero and
a
unit
whi'ch �beY£! the c anc ellatiC?n law and in which every element }.las � operty a ,
.
but which is non-as sociative and non-abeli an is given by
G •
{1, 2,3,4,5}
following
with
table o
multiplic ation
0
1
2
3
4
,
0
0
0
0
0
0
0
1
0
1
2
3
h
,
2
0
2
1
,
3
h
3 0
3
4
1
5
2'
4
0
4
,
2'
1
3
0
,
3
4
2
1
S'
1
We s ee
�{ 23) 4
.
Definition
is
an
�.
eleme nt
.J 2 {34). . and 2
-·
•
3"3
..
�
•
2
An algebraic nilpotent
3. 5.
x
-
..
-·
� is
such that
•
�f � groupoid
unique for s ome
n
and
with
tor
a zero
n
this
1
0 .
De.rini tio n 3 .6.
x
ment
such
�
that
We next . state
��a
�
�
�
�
3 .8 .
a
Let
An
•
0
to�
S
�
Suppo se
func ti o n
n
S
F
a
semi -group
is
an
ele­
•
to Fauc e t t
!. compact
Assume f'tlrther �
� there exi sts !.
some
r es ult due
� idempotents .
•
algebrai c nilpotent of
[3 ] . �
�-group �
!�
� !. �
!!_ irreducibly connected between 0
�S
c ontain s � �� algebraic nilpoten t& . ·
�
S
onto
I
,
.!:!!!. unit interval (01 1]
� !:!.!!, numbers !!.!:!! the usual topology � � � multipll:ation, � ,
!! !i! isomarphism !!. !!!!. !! !!!. order
Lemma
of
G
3.9.
Let
G
� .!
has property Cb and
- ---
G
p re
groupoid
s ervin g home omorphi sm.
� ! .!!!:2. � � every element
satisfies
contains � !!2,!!-.!!!:2, •lgebrait:. nilpotents .
the
c a ncellatio n law.
'lhen
G
Suppo se x!l
Proof.
or
we
rt-1
•
0
x •
se e th at
0
!hen xn-l
o
0 or xn- 2
X •
Similarly,
0
0
•
o
•
:x:
0
0
•
�
e
X
ntus
o
X •
1
0
Proceeding in this way,
•
The orem 3 . 2 .
G £!. !. compact groupoid
Let
which is irreducibly connected between 0
2!, G � property
and u
with
o
!.
� � !. unit
A s sume
� element
� � G satisfies the ·cancellation law�
(L
there exi sts a func tion
r
[ o, 1]
G onto I , the uni t interval
fr om
1hen
!!.!_ real numbers under the uaual topology and the usu al multipl�oation, which
!:!
� ii OJbo�hin ..!!
!!,!! !!
3
Thi s result follows immediately from Lemmas
Proor .
and 3.9.
Corollary J . J .
idempotent
.!!! order preserving h ome o morphism �
r
� G be .! compac t groupoid with
which is center associative .
nec ted between 0
and
f
G
1
•
.$,
3 . 1, .3.8.,
!. � and another
Assume G i s irreducibly
�­
satisfies the cancellation law, and every
element of G !!.!! pr ope rty (L
•
Then there exi sts
fun c ti on
a
f
from G
� I , the � interval [0, 1 ] � real numbers � the usual topo5 and �
usu al mul tiplic at�on, which is � isomorphism � well
.!!. �
order preserving homeomorphism.
Pr� .
G
Since
ro
Siml.larly,
Of
i s a connected set c ont aini n g
•
fG
•
t
act e :is a right unit on G.f
is
a
•
two eided unit on G
G
Since
•
and
•
t
and
0
f
,
·
is a center as soci ative idempotent
G and a left tmit on
fG
•
G
•
Hence
f
the result follows from Theorem 3 . 2 .
Remark. We note that any continuous multiplicati on on
[ 01 1 ] with
the usual topology i s the usual multiplication .for real numbers if the multiplic ation obeya the cancellation law,
unit,
an d
every element of
0
ae ts as a z er o ,
[0, 1 ] has property �
•
1
ac ts as a
29
EXample 3 . 2
with the usual
has
topology
c ellation law.
of real
[0, 1]
which is non-as soci ative and non-abelian, which
as a zero , which has
0
that
An example of a continuous multiplication on
We define s
as a ri ght unit, and which obeys the can­
1
ab 2 wh ere the last multiplic ation is
•
aob
numbers o
We next consider the as sociative-like properties pos se ssed by compae t groupoid s with a zero and a unit which
twe en
and
0
� G be
.3 . 10 .
� is irreducibly
law.,
cancellation
and
in
b
G
[0, a ]b
0
�
Now,
x
C
<
( Ga)b
Similarly,
[0, ab ]
a
•
•
Let
z
0
'-
C
[0, a ]b
a (b G)
( ab) G
nr
.
Thus
(ab) G
for all
a
€
(0, a ]b
z
'5. ab
Gx
is
a
•
!,! irred'ticibly
xG
a [O, x ]
Lemma 3 . 6 �
[01 a]b
•
We will next show that
'lhen
z
•
xb
by Corollary .3 . 2
by
(Ga)b
Lemma 3 . 6
where
and
z
E.
(Ga) b
and
be !. compac t groupoid wi th !. �
� Gx
•
.,
'lhus,
•
[0, ab ] .
G (ab)
•
=
•
0
[0, x ]
two sided ideal .
and u
and which
the
.--- sati sfies .. - -is ! � s i d e d ideal of G
•
(Gx) a
•
Also,
[0, x]a
xG
! compact groupoid
connec ted between
2 !. unit
C: [0, x ] � Gx by Corollary 3 . 2 ,
[O, ax ]
Similarly
� .3.12 " . If G !.!,
which
•
by Lemma 3 ., 2 .
[0, ab ] • G(ab)
Le t G
a( Ox)
�mm� 3 . 2, and
a (b G)
and
� irreduciblz connec ted between
Proof .
Gx .
G (ab)
C [0, a )b
Henc e,
cancellation law.
•
�
and
0
by virtue of Lemma 3 ., 6 .
G (ab)
Lemma .3 . 11 .
which
(Ga) b
[ 0, ab ]
•
[0, ab ]
connec ted between
'lhen
!!!2, ! �
u �� which s ati s fie s �
!. compac t groupoid with ! �
•
Proof.
:J
irreducibly connected be-
and which obey the cancellation law.
u
T.Mma
are
0
and
is
�
u
C [ 0, x ]
•
[0, xa ]
a
two sided ide al .
!. �
2 !. �
and which obeys the .2!!!,-
o
30
(Gx) G
cellation law
�
[0, x ) [0, u )
E::
z
0
�
t
z
e
[0, x]
(Gx) G
�
u
3 .13
Proof .
=
,
•
� !!!'
za
e
�
z a
�
0
X
for all
y � x
z
and
-
G (xG)
Thus ,
Henc e,
•
J3y the above method ,
-
•
•
and
[0, x ] [0, u ]
�
G
and
x . by Corollary 3 . 2
zu
e
( Gx) G
•
...
[0, x ]
•
•
G i s ! compact groupoi d wi th ! � and ! 'lmi t
�
b]
a [ O,
•
!.!
3 .14.
( GV) G
then
V
!!
G
a
all
0
and
u
�
which ob e ys the �­
a
and
b
in
G
•
[0, ab ] • [0, a )b • (aG)b
by Corollary 3 . 2,
!. compact groupoid � ! .� and .! unit which
G
z
€
G(v G) C G(VG)
(GV) G
(GvG) a
az .f; GVG
ideal of
.
.
C. .
G
�
for
•
G (VG)
a
of
If
Proof.
G (VG)
xt
co nn ec ted between
a ( Gb)
all sub s e ts
e
where
(0, x ]
0
is _irreducibly c onne cted between
z
two-sided � �
3 . 2, and Lemma 3 . 6 .
Lemma
tor
yt
-
a (Gb) • (aG)b
law,
=
[O, x ]
n
by virtue of Lenuna 3 . 6 .
G (xG)
If
� irreducibly
o ellation
[0, x ]
[0, x]
•
shown that
.. Lemma
Lemma
�
yt
�
0
z
If
•
it i s easily
which
a
is
G(xG)
w
[0, x] [O, u ]
Then z
•
Thus,
•
[0, x ) [ 0, u ]
•
(Gx) G
We first note that
Proor .
Let
and
.
x e o
all
G (x G)
•
If
C GvG
all
and
u
(GV) G
a
and which satisfies the --cancella-
-
G (VG)
is
a two
ideal of
sided
0
•
(Gv) G for so me
z E
( GV) G ,
by Lennna 3 . 12 and
(GV) G C G (VG)
z
GvG
E
GVG ,
C GVG
a
and
-
in G
z
€
for . all
•
Thus
a
GVG
for
some
€
v
V
Si mi l arly
•
v
V
i n G by Lemma 3 . 12 .
a
( ClV) G
•
G (VG)
Hence ,
•
is
Henc e ,
•
S�milal'ly1
a two
sided
•
Lemma 3 . 15 .
If
G
is
! c ompac t groupoid with ! �
�
!
�
whi ch is irreduoibly c o nnec ted between 0 � u � which s ati sfie s �
.
c ancellation !!!' the n GV and VG � � sided ideals of G for all sub-
31
se ts
V
G
of
Proof .
az
E.
a ( Gv
za
€
GV
•
If
C.. GV
z
C Qv C GV
)
for all
v�
s ame method,
€
a
€
z
,
Gv
for all
G
a
GV
and
for some
G
in
V
o
Henc e
3 .. ll o
by Lemma
S imi larly,
is a two-sided ideal of
�
G
i s a two-s ded ideal of
..
€
v
G
By
o
the
•
We next apply the above "as soci ative-like " propertie s to extend s ome
of the -results given
by
and Numa.kara..
[12 ]
Koch and Wallace
[16 ]
for mobs
to compac t groupoids with a zero and a uni t which are irreducibly c o n­
nected b e ·ween
0
and
De.fini ti o n
3. 7.
G
i de al of
J (A)
CG
A
If
J (A)
, let
and which ooey the c ancellation law.
,
a groupoid,
0
•
'Ihe orem
x
E..
-
� G
3.3.
J(A)
Sinee
A
Lemma
1)
V
!!!•
�oof .
•
is
Then , if
If
J (A)
'lhen
x
be
A
U YG U GV U
E. V
G
C J (A)
Theorem
A'
If
•
!.
c ompac t groupoid
A
'lhus
•
includes no i de al of
J (A)
C J (A)
lhen
0
3 ol4
i� c ontained in
be
!
J (A)
V
c·
A
about
J (A)
•
r/
0
.
Le t
by Lemma
Wallace
x
and Lemma
c ompact groupoid with
which i s irreducibly connected between
c anc ellation law.
Suppose
3 . t2 .
[18�
such that
3 .1$,
thi s se t
Therefore
is open.
G
Let
By Lennna
o
unit
i s ope�.
i s c ompact , we apply a lemma of
C .A
!.
and which s atis fies the
G
U GXG
�
!_ !!.!:2.
xG
and hence
3.4.
we are done .
U Ox
u
J (A)
i s open ,
,
and
�
U
GVG
and
0
t o produee an open s e t
3 .14
a nd Lemma
A
0
•
open and
is an ide al of
x
include s an
, the null set .
which is irreducibly connec ted between
cancellation
.A
and
be the union of all ideals c on tained in
i s the large s t ide al contained in
J (A)
G ,
u
\u
•
[0, u)
0
and
!!
u
!.
zer9 -
�
! unit
and which s ati sfies
the maximal proper
�
�al £!:_
G
•
32
ll
c ontains :very proper i de al 2!_
Proor . ·
If
a , 0 ,
th� � e sult is trivial o
a two sided id eal of
Thus,
A
C G\u
G o
a [ O, u)
Simi larly,
G
If
o
[0 , u) a
H{u)
• u
E
G' u
H (u)
is
•
If
[0, u)
Thus,
G ,
a • 0
u
�
H (u)
Clearly,
such that
an
1
is
S:. A o
G be !. compact groupoid � ! � � !. �
0
� u
i s any subgroup of
and
which obeY! the �­
�
G
c ontaining
E.
H (u) , there exi s t s an
u 1
•
Proof.
i
C:. [0, u)
i s any proper ideal of
A
which !! irreducibly eonnected between
If
by The orem 3 o lo
•
Corollary 3 . 3 .
cellation law.
u)
C: [0,
u
€
xi • u
H (u) o
•
If
id e al by Theorem 3.4.
If
x
€.
Thus,
x
G '\ u
1
H (u) • u
xi E.
o
G \u
si nce
IV
CHAP TER
APPLICATIONS
Monothetic Oroupoids
Definition h . l .
exi s ts
an
E G
a
a J and
A
De finition 4. 2 o
there exi s ts
in
e G
a
is dense
a
A group?id G
a
o
c ompac t m n o the ti c subgroupoid
Le t
f., G
a
A.
Since
•
A
c�llection
is
c alled a generator
G
of G
•
c ontains a
o
b e the c ollection of all
groupoid ,
is a
G [ 2] .
fi ni te Bwords "
of all
topological groupoid
A
there
G is monothetic iff
groupoid
a
iff
c an be expres sed as
c alled a gen erator of
element
�! c ompact
The orem 4 . 1 .
Proof .
is
topo� o gi c al
The
o
to be cyclic
every element of G
s uch that the
G
in
s aid
is
_
s u�h that
finite 1tword11 in
a
_
T
is a
finite "Words" in
groupoid, and
A
is dense
in T .
De�ni tion 4 • .3.
CLn
Let
There
•
a:re
lL.n1 , •n2 ,
•
·
a manner
bm+1 '
• • •
•
• •
for
·.
,
all
• •
a
groupoid and
,
a nN ( n)
��
•
'
• •
b1
•
,
•
'Ihus we
ob tai n
[ 8 .] .
&nm C ) ,
n
a1N ( 1. ) and
• . •
}
in this section
&nN{n) , to
po ss ib le sequences
ar e
the
•
this
lnu(n)
a ' l. 1
(n+ ) ,
• • •
independent of the
CL ,
n
the elemen ts of
{bns n
e
I}
•
element
,
of · G
letter a }
the se
{ �.N ( i) ,
se qu en c e
bm
an
Denote
ltelabel
•
if
are the terms
a
a
in
wor� s
_
'bm-tN (n+ l.)
The theorems
l.n2
•
•n '
2
that i f
,
G be
{c?ll�ction or all n letter
(2n - 2) s
N �n}
nl {n 1)! such words
&n ,
1
• ,
Let
ip
•
•
a:r.ry way by
a2N ( ) ,
:a
sequence in such
then the
terms
a (n+1) N (n+�) " .
way we assign
i .e . ,
�n1 ,
the theorems are
4.2.
Theorem
�bn�!l
�
Then the
Proot .
e.
there exis ts ·
n
m
E.
�
and
0
0
clusters
e
is an idempotent of
A
about
is dense i
Theorem
·,
groupoid
G
property
(2
�
n
:n
�
e,
and
e
•
bn
•
e
1
I
e , there exi sts
k
for all
{bn sn e
Th n
i.eo
g
for some
4 o .3 o !,
n
G
k
bk
I
Thus ,
o
nece s sary
�
e
0
{
r}
€
I
about
e
•
Then
a
and as sume
I}
�
G
�end-group, ·
P
p
of
U Q
and
G
{bt+J
-
oc)
··
1•1
e
•
sufficient c ondition for ! monothetie
G
2 .3 .
Henc e
•
CL
has property
{an & n
f,
1
}
I
i s a semi-group by Lemma
all the
G
•
2.1.
Clearly a
is
a
€
a
�
o
!?.!. ! monothetie groupoid � .! unit � sup­
where
P
� Q
� disj oint , closed ,
-
�
e ach c ontain infini tely manz finite words
Q"
Since
•
�­
�
!El.
G
b7
•
Let
1
G
�
4 . 4o
P
be a generator of
•
large st inte ger
a
such
clusters at
Since
P
and
G
o
Denote the unit of
Q" are open, each c ontains ele-
To establish the conclusion for
there is
p
But this c ontradic ts the
0
bn: n c
of
a
are unique by Lemma
Proof .
1
>
I
-
If the generator
·
1s
�� -mat· G
generator
1
•
'lhe orem
-
I
f,
p
k
such that
0
e.
clusters at
mono thetic topologic al semi-group has a generator wi th property
empty.
k
e , given any
o
a _ .
to be a monothetic semi-group [11 1 is that its generator have
- -
a
;}
ar
- - -
Proo:r.
powers or
G
Otherwi se , there exists an open set
�
� G
each idempot'ent
€.
r/ bt
�
that for any positive inte ger
fac t that
!!_
bm
such tha�
> k
and any open set
}
generator
From this it i s clear, if we are given any
o
N.�xt, . . we S'!ppo se
•
�
be a monothetic groupoid
I
Suppose
positive in te ger
e
G
Let
net in
k
Q
such that
o
But,
'btt €.
{bt+i}:.1
P
•
P , suppose
'!hen the net
clusters at
1 by
Thus
Q i s closed and we have a con tradie­
·
T o establish the concl us io n for Q , suppose there exists a largest
!heorem 4. 2 .
tion.
-
....
-·
k
inte ger
an�
-
s i nce
1 E. Q
..
.
such that
clusters at
1
E.
�
Henc e
•
but the latter net is in
Q
{��-1 i s a net in
{'b)cbtc+i}Cl0i• 1 cluste s at bk .
The n the net
•
by' c ontinuity
P
E
btc
Thus
•
r
s i nce
P
P
Theor�� , J.t- . 5'.
ia a closed ideal
- -
Proof,
Q
of
Let -a
of
V (b)
a
�ere erl � t b j
•
e
Q
n E
__ _
.
I -·
e
of
•
to � � �me
V(a)
a�l p�sitive integers
'}
� n
_
De � ni �ion
of _ 0 .
_
_, · Lemma
. .
Q1.£b1 ti
impli e s
4.1.
�
A C:
Proof' ,
j
_€
I
4. 4!.
If
.
n}
CD
Let .
�
n
Q1 {bi �1 . .
2..
·n}
We f'!rs t assume
C
e
b +ny
n
n}
V(ab)
a
·
V(a)
But
•
for all
V(b)
bn+�E
far
V(ab)
Q .
is a left ideal oi'
B
a subse t
is
letter words with lett er s in
•
V (ab)
Clearly,
•
!. c ompac t groupoid with !
0 be
Suppose
•
•
• _Q1on g � only i!
00
bj
Hence
i s a groupoid and
G
}.
'bn+Dy
and
a
G
n
z.
V(a) V (b)
Q [bi ai
C\ {bi i � ij
'lhus
deno te s the totality or
..
Q. fbiai· � n}
�en there exist neighborhoods
ab E
n and
Similarly, it i s right ide al or
1fl
ab
C\. {bi•l
n y- i s a pos i ti ve inte �r -
��·�
Q1 {�a i
and b
b respectively such that
and
E
we
•
ll G !J! ! c ompac t monothetic groupoid
is an arbitrar.y nei ghb orhood
and
,
·
i s closed and
hava a contradiction.
P
.
A
C 0 ,
----bi l i 2:
Cl, {
n}
B
� and
A
•
A
,
of
G
,
•
a
�
�.G. Then
AA
•
A
A •
A!
36
CD
n
rl G
n•1
�
�
Sinc e
•
is a two sided ideal,
A
'lhen there exists
!
i e c ompact and
x £
2, a•(/t :J G 2m
x
A
A
C
Then
k
Let
Since
x • bk
�
n}
a
sueh that
, en)
"V(a n)
!(V(c1} ,
E
N m_r,
•
be
a
.
No
,.,
c1
n
x e, G
{bn}
T(c1) ,
£
br
a
W_�br1 1 .bra'
c1, c2,
G
n
• • •
• • •
,
c1,
,
A
Thus,
•
r1}
�
•
n
•
•
,
en
e
X
--;y
{b i t i �
o
{bnt D
since
�� ,
n}
• • •
en ,
, en
in
G
We denote
Suppose
o
� I}
• G
such that
bk
ci E.
•
by
{b ii 7 n} .. .
b e an arbitra.I7 neighborhood or
, , and
V (e2 )
e
•
Therefore there exis ts
c1, c 2 ,
e2 ,
n}
We will first show. that
o
,
{bi i ·�
E
�
bn,.· n., , o • • ,
,en) )
• • •
AA •
{bit i
nl
n•
.
and suppose
k <:.
where
en
C V{W{e1, ca,
v(o n) )
• • •,
•
Thus, as asserted above no
ments
generator or
Ci
en) )
c2,
1
Hence
•
V{c1) , V(c 2 ),p , •
lhen there exists neighborhoode
of
W (el, ea,
C V
imPli es A C Q {bi t i �
A".
Let
x • bk
•
n•1
such that
it may easil)" be
CIO
is a word w1 th letters
"'V(W{c1, c 2 ,
'br1
AA
,.
.
• • •
na
Hence
G
G i s !· compac t �thetie groupoid, then
•
. . .,
aod
n} �
x � a n , there exist elements
-· ..
m
G
Sinc e
•
� A
V
have a contradiction and
we
is not a cluster point or
W(e1, ca,
Let
"?-
If
S
I�
•
{b1 t i
x
"
A
,
. Proof .
£ I
'
n• 1
The orem 4 . 6 .
0
an C
Hence,
•
[btli
n
A . n- 1 rf1 c
AA
E.
x
ar gument1
such that
m
.
.
Suppose
o
is closed, '\here exists an open se t
A
shown there exi s ts an inte ger
X
such that
A
B VV [18.) • By USe of an elementary net
X
·c A
AA
•
respectively such that
•
en) )
, , .P
{bi t i �
•
n}
bt
bi
where
where
and
ct e., {bi a i �
is a
But there exi sts
o
, . , bn+ny f, V{ c i )
brn)
• ,
t
o o
o1
� n
br E..
n
o
V(c n)
•
Renee
have a contradi cti on.
we
n}
1
Thus , each o r the ele­
•
i < n
•
Therefore
bk • bp
··
p � n
where
k.
'::;>
btc e
Thus,
�--�}
{bi 81
�
n}
and we
37
have
a c on tr a­
diction . Hence an C
bt i i
fpr all n and
C r'\ an •
� n1
As above,
on C ;; b1 t i � n
1
n•1
'�
n
1
h-1 � �--------�
n•1
�
� n} " Hence A C G , A • A , and AA • A
'lhus, ()1G0 • (l1
n• \.:
ni mpli es A C
...
by L emma 4 1
n.{bi li
o
·
fb1 ti
Q {b.-1-�-�--n}...
i s a monotheti c semi-group,
If G
·
1
sided ideal of
•
Koch
An
Example h. l .
s emi-group is
is
the minill"..a.l two
In this special case, Theorems 4. 2, h.4, and 4 . 6
G [19 ] .
have been proved by
eye lis
..
Q (bi : 1 � �}
G
( [10 ]
example
and
[111 �ection 3 ] )
of
a finite cyclic groupoid which is not
1, 2, 3
a
a
wi th the following multiplication
..
table .
Where
X
may
be
of
G
generator
2,
1,
�
,j
iff
and
generator.
i
2
1 2
.3
3
X
2
3
X
1
3
X
X
1
or
3
��
ClearIT,.
�t;
and
(12) 3 , 1 (23)
1
a
is
•
An
Example 4. 2 .
&t • •j
1
•
j
•
example of an infini te cyclic groupoid which is not
Define the following multiplication&
&t•i • &t+1
•
Clearly
(a1 a2) a3
, a1 (a2a3)
and
a ia j
•1
=
a1 if
is
a
If we use the discrete topolo gy in Example 4.1 and Example 4. ? we
obtain monotheti-c groupoids which
-
Example h • .3 .
are
.
n_ot monothetie semi-groups .
Consider the interval
[0, 1 ]
of real numbers under
the ueual topology
mu�tipli c ati on on the
tl?n'� .
�e !J.
!��� - is not
potent
then
_
-
•
•
..
A
_
J
�
We
e, .
a
the following multiplications
and
I}
right is the usual
note that 1�
C 0
A
...
�-
S: . Q{bi ti '?- n}
ple 4.�. C onsid er the
l!Xam
They form
a
{bna n
•
t,
•·
t
e !eme�t �r
G
l , · and
AA
•
A
1
•
integers with the usual topology 1mder
a gen erat or
G be
Let
a
of this
A
groupoid .
Zero
topologic al groupoid with a zero and
such that an
�
•
monothetic topological groupoid which is not a
. One is
De.fini tion 4 . S .
an
a
Let
vi th generator
I} , A
£
Topological Groupoi d s With
be
ab 2 where �e
•
real multiplication.
is a compact mono theti c gr oupoid
topological s emi - gr oup
a
b
monothetic semi -group and which has zero . as its only idem-
�- -�
�ubtra�.�ion .
ao
is unique .for all
n
!
If . an � 0 ,
i . e . , if for e'!e:ry neighborhood U of 0 , there exists a positive inte ger
� • �(U, a) such that an € U for all n � m , then a i s termed a
nilpotent element or
�fini tion 4.6.
groupoid
G
with
consists e.ntirely
or
G is
a
sh�t
in
a ni lpotent.
!. !!!,! left (ript, � sided) !!!!!.! or
a z ero
is
a
{ri ght,
left
!!!,! semi-�
A
semi-group (sub-groupoid) of
0
of nilpotent
potent e lem ents .
Lelllfta h. 2o
�
{A�
a
X
E. /\}
sided) nil ideals or ! topologic al
sided) ideal of
two
elements ,
_
.fam:i lz
�
whi ch
{sub-grou.poid )
of nil-
� riglit (lett, �­
! �·
Then
A
•
hl A ).
a right (left, two-s id e d) nil id eal of o .
r--' - -Theorem h. 7o If G i s � elementwise bi c ompact sroupoid with
ie- also
--
G
consisting entirely
be !.
gr oupoi d
a t opolo gic �
!!:_21
-
�very
-
....._._
- --
right (left, �-sided)
�
-
A
of
!
G is either ! nil id e al
tains
� con
id-.potente prol'ided
�-.!!!:!
3: tL : ·
Prt:10t. Let
A
be a
o.. in
A
��
�;�e� t .. a
r/
39
every
element �
non-nil right ide al of
that an .,L+- 0 si nc e
G
•
tar:
A
h!! pro:per-
-:then there
all a E. A
is
,
an
the
powers . .t>! a are unique by Lem.ma
2 . ) . N ow, it we consider subsets
\·� {&n�.Jl e:, l} , !y • {ai i � v} ( V • 1., 2, 3, : ) , and
D /'4 B; !
.?'. is a sub group of a and BD c.;;. D b7 7heorem 2 .k. Let
be the identity or n Ir e • ·o D (0) and an--+- o � . But.,
�is !�n�adicts the as81211lpti that . an 11 ,.... 0
'!hue
e is a non-sero
.
•
ideapotent.
Since BD c.=. D ,
ae e. D .. and..
denote by (ae). 1 the in•
.....
.
nree ot ae in 'D Since A i a a right ide al, e ·• (ae) (ae) -1 E . J.
H�me A
It i a the s ame vi th the
. e�tain�.. a �on-zero idempotent e
or �he lett ideal.
�
1heorem· 4.8. ! closed subgronpoid CJs. � !! elementwise bioomj>ac t
voupoid � ! .!!!:2 !! ei tb:er ! � subgroup id � contains .�-.!.!::! idem­
pot4inti j;ro"Vided each element !! � has propertj � :
ProOf. nte existence .�t· · the non-tter o i �empotent e i s established
as in the �o� C?t Theoren 4. �. . On� t�en n o te s that
e C D C · tf� 01
De�nition k . 7 � topolog1cal . groupoid wi th a zero i s said to be
.
an N •groupoid it 1 te nilpotent el e�n �s form an open set.
L- k . J. � G � !. topological gro poi d � !. � � !.!.! a
� .!!!. el nt or � � p�pertz a . !! an !! ! nilpotent elem�tlt.
:rc,r· iome poai tiye integer n , theh !' itaelt !!_ !. nilpotent eleJiel'lt.
Proot. All the powers ot a are unique b)" Lemma 2 .3. Let b aD
and let tJ be an
bi trary neighborhood ot 0 ; !hen there exist n•igh-'borhoods Vi (1 • l, !,
, n - ) ot zero such that W1a1 C lJ
1
t
·•
•_.
.
• •
_
'
e
,
•
•
on
1
•
we
I
.. ..
I•-'
,. ,,
•
..
•
'"'
•
"'
.
•
.
c ase
o
_
•
.
•..
.
u
eme
.
'
•
•
...J
ar•
•
�
.
• • •
••
.
•
(i
•
1, 2,
!\
(i
• • •
- 1)
, n
1, 2,
=
•
40
, , n
•
Now, sinee b .,.. � 0
o
such tha t
1)
-
M*
Let
•
If
� � M . 1 we vri te
k
�
_.,
�
0
•
I
we
+
i
have
e
•
a
G
C: 0
•
1
If ·
Ah �-lA
•
• • •_,
1heorem 4.9.
o .
o'
Mn_-1)
0 :=. i � n
aU
m
akn
Wi ai
M
and let
b�
•
C U
e
u
•
6
etc o
1
G 2! !. topologic al groupoid with ! 0 in
Let
!,.! G !!.!!
•
!, nei ghborhood
is an element �f N , there exists a positive inte ger
p�
E
V0
a
U(p)
.
of p
q
an
��-
4.10.
·The orem
An
element b
Proof.
G
and
p
Let
,
u (p)
E.
..
x(yb)
of
a
C N
•
that there
a
•
If
nilpotent.
Hence,
•
G
a topological
•
groupoid
for all
is
N
such that
C: V0
q"fl
of
V0
x
and
G is said to be
G
y �n
[14] .
'l'he radical 2,! .! bioom.paet N-groupoid !! � E!:2.­
vided each ,nilpotent element
€
E: -v0
qD
a zero i s called the r �dieal or G
right a ssoe i ati ve iff (:x;y)b
x�
� V(pn)
1he j oin o f all left nil ideals of
'Defi niti �n 4.10.
any
U (p) n
·
Definition 4 . 9 �
w1 th
=
"
4� .3 and
s _ a. nilpotent by Lemma
N-gr�upoi�.
groupoid
�
-
_ _
/'-.
such that
any element of U(p) , then cf
is
n
Recallin g the abo-ve convention,
one can e asily see
-
neighborhood
1hua�.
is
, 0 , we
which consi sts en �ire!z 2.f nilpotents, �en G is � N-groupoid.
_
Proof . We denote the set of all nilpotent elements of G by
•
•
is
p
q·
1)
a is a nilpotent .
If
is
+
Then
If i
•
Henc e
•
n (M*
=
1 [18 ].
-
Wi
Mo implies
Y �
1
which every element has property (l.,
.
impli es b ¥" €
1
� �A
0
Mi.
a topological groupoid and
"
� "'AA
, we shall use the following notation t A • A
A
Definition 4.8 .
A
kn
•
akn+i aknai bkai
•
aU
have
i
- If'
•
max(Mo, M1 ,
=
there exist inte gers
Mo such that
and there exists a positive illteger
bv � U
�
"V
:J
R
·x�a
!.!. right assoeiative o
be the radical of
C R C N
,
where
G
N
�
and let
a
€
R
o
Then far
i s the set or all nilpo te nt
G
elements of
�d e�l
The above re lati on is valid since
•
��re exi sts a ne i ghb orhood
x�
f_or each
�d
-�
Vl\(a)
€.
1
t
•
GJ
1, 2,
v,�
GV(a) C
U
V(a)
•
k
. . ..
G ,
. of
,
right associative .
Thus,
· Example h..S.
on [0, l ] s a o b
•
ab
G is
of x:A
:tJ.vCk""-)
, ·1 .. 1 , 2,
·v(a)
o o .,
k
.
C
·
•
G
•
QvA1 (a)
'lhus
•
Let
2
t
*
C R
�
md
..
-
u
a, b E.
is_ �ormed in wh i ch
But,
·
Define the following multipli c ation
-
�ot � N-gr �� oid .
is open.
be the close d unit interval of real
[0, 1 ]
for all
C
[0, 1]
where the multiplication
on the right
is the usual multiplication of . real numbers .
..
groupoi d G
N
Sin c e G is
•
so that
such th�t
a
V (a)
e.,
a
numbers under the usual topology.
•
C
C i-(V.Cx�1a) C N • If we put L* • GV ( a)
.
i s . a left nil i de al since e ach nilpotent eleme�t is
*
t
then.
V(xA )
C V(x� a)
a}
&
(x" ) IX,_
'V(XA 1)
Mv(�� i ) 'V(a�
T(xAa)
one can .f'lnd a f'inite sube ollee
ti on
k
{v
corresponds tok
1_ (k)
such that
V(x)\)V"(a)
neighborhood of
a
.
}
_
i s open by assumpti on,
find peighborhoods
ean
such that
U V(x�)
X�tG �
�e den o� by V ( a)
where
a
N
x).. a
of
-v�xAa)
Hence, one
•
of
N
bi c omp ac t and
{Ycx>.i)
4.2 . Si nce
G by �rt�e or Lemma
or
is a left nil
R
0
it
A
topological
is the only n:i �po tent element .
we
consider
[0, 1 ]
with the
He nc e ,
discrete
topology and th ! ab ove multiplic ation, we form a topo lo gi c al groupoid
G'
which i s a_� N-groupoid.
We next
c on s i der bri etl:r the concept of boundedness in topolo gic al
..
..
groupoids
w1 th a zero .
Derlniti- on 4 . .11. A subse t B of a top ologi c al groupoid with a
.
zero i � ri ght bounded iff tor 8!JY neighborho�d U of � there exis ts a
neighborhood
..
V
or
:1.�17 �etined, and
bounded.
a
0
such that
·-
· VB
C
U
•
tert boundedness i s simi-
s e t is bounded iff it is ri ght bounded and left
! bicompact subset �
Theorem 4.11.
. h2
�
! topological groupoid
!
·!,!!! is bounded .
Proof .
B
Let
be a bieompae t subset of
bitr&l")" neighborhood or
Since
B
Let
•
W (O)
Hence
4Cx)
i s bican:pae t and
tion {vC�_)
: ki
u
1, 2,
Qwx1 (0)
W(O)B C: U(O)
b ounde d .
We finally
Si nc e
•
O:x:
:x: £ B
k}
• • • ,
"! or
Wx1(o)
where
B
and
.:J
is
0
•
of x and Wx (O)
V(x)
nei ghborhoods
0
G
for
or
0
x
U(O)
Let
•
€
B
be
an ar-
there exi s t
,
such that Wx (O)V (:x:) C U(O) .
, there exi sts a finite subcolleo-
B
{v(x) :x
}
t.
B
such
Simi larly,
ri ght bounded .
c on sid er the special c ase of
a
:k{v<xt> �
that
V(xi)
corresponds to
k
, i
B
1, 2,
•
B.
• • •,
k
is left
bic omp ac t groupoid with a
zero and a unit which is irreducibly connected b e twee n
0
u
and
and which
obeys the c ancellation law.
Theorem 4 .12.
which
�
!. bicompact
groupoi d with
!. �
�
!.
�
!.!!, irr e ducibly connected between 0 � u � � obeys _!:!!! cancel•
lat:ion law�
'!hen
--- -
-
[0! x)
(1)'
·-
i s ! nil i de al of
prope rty
(�)
�
...
(3)
property
virtue of
""'
�im:l1arly,
� ideal
(1)
a �
Let
..
G
J
e.
z
1heorem 3 . 1 and
[01 x) �
provide d !;!:! y
r/
X
� G provi de d !!!._
4
U
E. (O,x)
have
..
-
C [O, x)
and
By Lemma 3 . 10
[0, x)
Corollary
.
E..
(0, x] have
o
•
3 . 2o
for all
[01 x)
y
•
� radic al � G !.!._ open
sided ideal of
(2)
G
a.
[ o, x ] !!!.. !.
Proof .
..
G
Let
If
a .j 0 ,
If
a
•
a €
G
o
is a nil
0
,
C [O,x)
If
by
this result is tri vi al .
Thus , [0, x)
i deal by
[ 0, x] i s an ideal.
a[O,x)
is a
C oroll ar y 3 . 2 •
x .j u , [0, x]
two-
is a
•
43
nil ideal by Corollary 3 . 2 .
Le t
(3)
potents of
such that
L*
• 1
a e
R
and
C N
Hence,
t*
and
G
one may find
,
GV(a)
is
U V(a)
GV(a)
Lemma 3 . 1.4 .
__ ....�e
If
•
T{a)
1
- .
'!hen
G
be the radic al of
R
nei ghborhood
•
a
of
V(a)
a
as in the proof of 'lheorem h .lo.
C N
and
be the set of nil­
N
left nil i deal of
a
C R
a
let
by virtue of
G
e, Y(a) C R
note that i f every element of
G
Thus ,
•
has property CZ
R
,
is open.
i s an
G
N-semigroup .
Definition 4 . 12 .
A topological
groupoid
ha� arbitrarily small ideal nei ghborhoods
of-
U
0 c.. ontain s
-·
-
an
�
G wi th
0
iff
a z
ero i s said
every neighborhood
ide al which is a nei ghborhood of
0
'!his i s
•
Chapter 3 ]
analogous to the notion of arbitrarily small subgroups [ 1.3,
. .
_
.
which
_
.
.!!
o.
..
..
.
If
!!_!,
Proof . __
then
two e le� ts
!h�
ax •
b
a
and
groupoid
A
a!_ld exactly o�e
such that
ax •
Definition 4 . 15.
iff tor any two elements
,
�
y
a
b
and
.
� G
b
[7 , p.
A groupoid
a
and
G
obep
� �-
is called a quasi group iff for any
there exists exactly one
A groupoid
irr for any two elements
G
and
Quasi Groups
b in G
.
E.
0 .!.1!!. u
� ! .!!!:2 � .! �
result follows immedi ately from Lemma ) . 10 .
, , D�finition 4.14.
x
i s !. compact voupoi d
•
arbitrarily smi.ll bie ompaet ideal neigb.borhooda
0 !!.!!,
.Definition 4 . 1.3 �
that
G_
irre�c ibly c onnected be tween
cellation
or
I
'lheorem 4 . 13 .
_
to
b
s uch that
0 i s � alled
in
G
,
ya
a
in
i s � alled.
G
,
b
E 0 such
[ 7, p .
986 ] .
� � quasi �
there exists exactly one
986 ] .
0
•
:X:
a
right
� guasi
group
there exists exactly one
e
y
G
b
ya •
such that
Definition h. 16 .
ax
lation law i ff
. 44
[ 7] .
G
A groupoid
• ay
imp lies
X •
said
is
X
y for all
G satisries the right cancellation .!!! iff xa
s �ti sfy
to
y
,
the �
and a S
1
•
ya implies x
u
cancel­
G J
and
y for all
G s at i s fi e s the c ancellation law iff it satisfies
x. , y ,.- and a � G
the ri ght cancellati on law and the left cancellation law.
o
We
now
state a well
lm own
theorem from the theory of quasi groups
• .
!_ finite cancellatio�_ groupoid is a quasi_ group • [ 7]
1'heorem 4 . 1.4.
__
Under certain additional hypotheses, we will extend
the
the or em to bi-
compact groupoids .
� G be
Lemma 4.4.
�ubset £! G
uJ b
�d
W}
an�
B
(�) be
•
!!!1 W
•
! topologlcal groupoid
[bc&)8bw
•
!!!.
index
G
�
system
!!!.<!
w
� B* be !. bic omp act
.!!!!!. �
e,
� � _e., r � Then, there exists b
c
{awtwt w�
•
a
•
Let At
•
•
of
•
B
Let
family or subse ts of B*
B t1 ,
• . •
�.
�t2
.'
�tn are
borh ood
n e � gh
Let
• • •
Ato
•
{vt(a) & t
Vt (a) n
the set of elements
of elements of At
•
/\.
Let
Proof.
of
_w}
€1
A
• .
�
€-
T}
Clearly,
whose elements
B
..
{Bt g t
E,
wi th the fi nite
correspondin g
Vt0 (a)
Vt 0 (a) (l
resp onds to Ate,
•
(€
/\
,
�t
,; D
have the
T}
T)
<2..-
�
�
By
•
Bt , we denote
same indices
with tho se
We will show that
o
o
B
o
is
B
a
For let
At 1
!h en
1
At 2 ,
Hence, there e4l_sts a
such that Vt0(a) C
Qvtt (a)
deno te by Bt0 the subset of B which
n
_
(\
Then, 1 t is clear th at q 1 Bt0 C 1•1Bti
A
C B*
B
suppo se
intersection property.
subse ts of A
t0 E...
f., G
be a complete nei ghborhood system
, B tn be any finite number of sets in
the
We
•
� � ab
lJ
w
� subsets � G .
w}
whose elements correspond � � � � system W
{a.wi a
A •
and·
•
cor-
•
4S
B
Thus
has
t� -/
we have
0
neighborhood of ab o
Y(b)
such
or
b
b
·e
Since
Q�
C2,
property
n
hand quasi
n
Proof .
CD .
:
-/
r•1
.
0
E,
I
If
p
. •
Q�
CD
•
for
p
•
o
{
r•1
•
qG
As
•
r
•
1, 21
•
A
"
•
element
G
!!!!.
is .!
·
-- p
for all
•
i,
piG
'lhua,
E,
o . o
}
� r
o
Le
e.,
G
:J p 2Q ::J
'lhen, since
o
�
qx(x e
'
r • 1, 2,
,
let qx
;T •
e:
00
G)
•
p9
o
•
Th en
•
But,
(\
�G
r•1
C
qO
G
o
Let
there
is
a
g
Hence ,
G1
€
let
p'
(b
€
a
ce...
show that
and
and
C\,PiG
• {gi i e. I}
lfr
p 1 • qg
N ext , we will
pG)
Let
o
pr+1 ra .
f'\ prG
r• 1
•
or
one can find
But , pr+irx E
qx €.
q {pO) (x E
•
any eleme nt
be
€.
�
Lennna 4 o 41
•
f"'\p�Q • qG
i•i
o
Qpi a
where
is bieompac t
a n on-ne gative inte ger .
_
show that
Qpfa
CZ)
-
G
is
• • 0
'nlerefore ,
o ., .
I}
ir
where
0 0 .
o
o By
� •
:J pG
G
that V (q) :i: C V (qx)
such
1, 2.,
2,
that
arbi trary neighborhood of qx
we will
E.t
b e fo re ,
V(b)
Hen c e
•
�
We intend to sh ow that
•
(:l
q
of
Next
i
(pllo) • (ppn) G pn{pG)
{� ii
_,
qG C
I.}
{fcta i
pi z i
C
•
pig1 ,
qg E
q·{pO)
r
.
T) .
0
"
follows
it
() P;
be an
for
n ia _
i• 1
'!hen p '
�d · P �.
that
G 1
for
Q:)
-�
qG
is !. bi c ompac t · groupoid !!! which every
q e,
(
Bt · n
n Vk (a)
A
a
e..
k
1
€
G
Put P r •
V(qx)
€ V{q)
{a
Ak
'bt
o
and
€.
a neighborhood V q )
::>
we le t
and
•
•
Le t
•
, and let
tr�-
ako €
th at
a
( E /\
where Vk (a)
N ow, if
(I
ab e, �
of
neighborhoods Vk(a)
c
group provlded p
First, we show
·
· qG
.
plla :J
(\ J5;
0
be an arbitrary
V(ab)
�- vhich sati sfie!_ the left c anc ell atio n l aw,
!!!! f or_ !11
::>
-/
,
le t
md
C V( ab )
Vk (a) V(b)
. Lemma 4.So . !!
_ _
,!!!:!.
Th e n there exi s t
that
•ko ' bto e. -v(ab )
�
Q�
n
b
-v(b) n BJc
.
an element •ko su�h
there exi s ts
.
Let
o
i s bicompact
B*
the finite intersection property, and since
C
G
such
qG �d
r.lpi (pa)
a>
Y(qx)
�
be
.
an
•
qx
arbitrary nei ghborhood of
such that V(q) x
ot q
r
1, 2,
•
pr+i:l!'x
ntus
E.
• • •
rirr (pG)
pn (pG
)
.
•
we
show that
P'
•
, i
{!i
Clpi (pa)
•
1, 2,
pn+lQ
•
�+ir+la C
•
'lhus,
•
Q_JI' (pO) C\.pl'(pO)
qx _e
pi gi '
C V(qx)
S i�c e
•
one can find a
Then,
o
and
for r
p1
e:
pG
exists g '
e
o
•
2,
1,
a.pi(pG)
q(pG) C
gi '
, where
pr (pG)
Let
for
V(qx)
positive inte gers
.
•
neighborhood V(q)
E
pr+irx
for all
pr+lo
C q(pO) ,
• • •
46
•
n ,
o . .
•
Finally,
Q.p1(pa) Then
!Jet
I}
• {p1 i
•
P
f.
:c
¥ c per pG such
e I} '.nlen there
that
{i11.1. t i e_ I} " ;.- p 1 :thus, p 1 •
q(pG)
But, Qrcpa)
Henc e , Qpi (pa) • q(pa)
�d Qpi (pa � C q (pa)
Qpi+lo Qpia Thus, q(pG) qG and pG by virtue or the
�d
Qt
.•
qg 1
:i
I
0
E
•
1
...
E
Ill
qg1
•
•
•
•
there exiets
x
•
has property
G
�
lation law, this
Lemma
•
•
lett cancellation law .
--...&.
'lhus, if
such that
G
ie
b
ax •
G
and
a
---
1
a
left hand
quasi:.group.
every
element
the ri gh t c ancellation law, the n
----
group 'f!ovide d
p €. G � � !!!, n � I .
�. "!he proof of thi s
G
b in
B y virtue of the left cancel-
•
is
any
G
!:. bicompact groupoid � �
Q, and which s atisfie s
!!_ !_ right � quasi
•
we are given
x i s unique and
4.6. !_!
€
�
A
A
(QpD) p • G (p llp)
lemma
is
an al
-
.
•
ogous to
A
( Op) pD
�
G
� !!!_
the proof
of the
previ ous lenma.
Theorem
!l!!. pr operty
4 . 15.
fZ ,!lli!. which
gua�� group pr ovi de d
•
(pp'h) G
for
Proof.
If G !!_ !. bic ompac t
all
s ati s fi e s �
(Gpn) p
A
p € G
m
/\
( Op) pD
•
and for all
groupoid
!!! � every element
c anc e ll ati o n law,
G (ppD)
A
n
� I
and
� G !!
A
/\
n
p (pDG) - p (pG)
!.
•
'!'his result follows immediately .f'rom
Lemma 4 o .$
and Lemma 4 o 6 o
Example h o 6 .
�� er thi �
47
Consider the following multiplic ation table
multiplication
1
1 1
3
2
3
2
3
2
G
)
1
1
{1,
•
2
3}
2,
is a finite groupoid which satl s
fi�e the � an� ellation law, in which every element has proper
vhich
£
p
(Op�)p
G
Example
Define
�
a_
(CJp)�
•
.
M.d all
n
4. 7.
O�p;n)
a
e.
I
Le t
E
not a
.
'
semi -gr-oup
Ex�mple
...
. ..
4.8 .
Let
C
12
C
·
z
E
(xx
•
i� the d� �ter througn
4.9.
Let
P
relative topologyo
1'
and that the line
x)
�
x
•
a
x o
y
o
•
topology.
midpoint or the segment
r
y
be the parabola
On P,
yz
• z_
if
z_
ie
with
the �r
C
the ·
�ge
becomes a
emi-group although it is a quasi- � oup .
.�
define
(7
x o y •
z
•
x2)
in E 2
a�thougn it
is
wi th the
by the demand that
is paraliel to the tangent at
P
the
x
Under this multiplic ation
Under this multiplication
whieh is not a semi-group
For
usual
becomes a topologic al groupoid which is
define
topological � ou� id which is not
usual
semi-group.
is not" a
by defining
·and in
£or all
be the circumference of a circle in
On
__
Example
(p�) G
(l, ,
•Ithough it ie a quasi group .
ueual relative topolo gy .
y
•
be the Euclidean plane vith the
Under this multiplic ati on
•
ot
2
E2
�ultiplio �tion on
Q
But,
•
p(p1Ja)• .;n(pG)
and
tY
­
x
( set
becomes a topologic al groupoid
a
quasi group .
algebraic counterparts of :these example s, see
[2�,
p.
22h·] .
BIBLIOGRAPHY
BIBLIO CRAPHY
. 1.
•.
\2.
3.
Albert, A . A . "Quasi �oups 1 I ". Trans actions Ameri c an Mathema tic al
54 {1943) , PP • 507-519 •
..
_ S o c iety, Vol .
Etherin gton, I . M. H. 11Groupoids w:l th addit i ve endomorphisms 1 "
Americ an Mathematical Monthly, Vol. 65 (1958) 1 pp . 596-601 .
Fauc e tt, W .
Proceedin
idempotents . "
Vol . 6. (1955) pp.
h. .
Frink,
5.
Oarrieon1
6.
o.
matical
pp .
.
9.
•
eelf-di stributi ve
� 62' (1955) ,
G. N.
WQuaai- groups . "
4 74-487.
.
Haus:ni�n, - B
n al of
Jac obson, N .
N ostrand,
Kelley,
J.
10 .
K och, R :
11.
Koc h , R . J;
sity,
.
_
.A. . ·and·· o .
Ore .
, _
J.
Americ an Mathe­
Ma themati c s , Vol . 41 (1940) ,
Elementary Topolo gy•
1951.
York t
John
Americ an
J our­
N�A
.
"'1heoey o f
Quasi-group s , "
pp .
Lectures in Abstract Ali!bra,
Inc . ,
983-1004 .
I.
New
York !
D . V an
-
Ge n9ral T opology.
L.
1955 .
Annals of
Mathem ati c e yol . 59 (1937) ,
•
eyste:m8 , •�
69 7-707 o
pp .
.
Hall . ... D . W. and o . · L-. Spencer .
and Sons , Inc . , 195$.
_
a.
141-14
"Stimmetric and
Monthly, Vol
�
7.
r.
two
of the Americ an Mathematical Society,
"Compact s emi groups irreducibly c onne cted be tween
M.
"On topol o gi c al
195!.
New York s
D . V an N ostrand, Inc . ,
semi groups . " Dissertati on, Tulane
Um ver­
•on monothetic semigrOu.ps . w P roc ee din gs ··of the Ame ri c an
Mathematic al Society, "Vol . 8 , J�57) , · pp • ., 397-401 .
12 .
Koeh, R. .
13 .
Montgomery, D � and--L . Zippin. Iopologieal 'l'ransfonnati on Groups o
.
Inte rs eie nce Publi shers, Inc o , 1955 .
York 3
J . and A . D . Wallace , "Maxi mal i deals in compac t semi groups . •
Duke �1athematical Jour nal , Vol. 21 (1954) , pp . 681-685.
lh.
Pios tert,- -1' . s . and A . L . Shi elda .
•On
a c omp act mani fold wi th boundar.y. "
(1957) , pp . �17-�3 .
lS .
Nnmakara,
16 .
lhnnukar a; K.
Yumagata
the structure of s emi groups on
Annals of Mathematics, Vol . 65
b!collpact semigi-oupa ."
X.
"'n
Okayama Uni versi t;r,
Vol . l (19 52) ,
pp .
Mathemat:t e al JOU.rnal
99-ioB.
"11Qn bic omp aet send. groups with a - zero .
Uni-versity,
N atural S ci ence ) , Vol o 4
(
New
.
"
o:r
-�
Bulletin of
(1951) ,
pp .
46S-4lf.
17.
Rees, D .
" On eemi gr oup s o " Proc ee dings of
sophieal Society Vol. 'J6 (I94o), pp .
�
t�e Cambridge
398-460.
Philoao-
.
_
16 .
Wallaee , - A . D.
'ItA note on aob s, I . " Anais da Ac ademia Bruileira da
Ci enci aa, Vol . 24 (1952) , PP o 329-J34 o
19 .
Wallace , A. - D o
•A note on mobs 1 II o • Anaie da Ac ademia Brasileira
da Cieneias , Vol . 25 (195�) , pp . 335=336.
20 �
Wallaee , A . D.
An Outline for Algebraic Topologz, I .
a i ty, 1952 .
n..
Wilder, R. L.
2�.
2�3§.
:
of Mani folds . •
'!he Americ an Mathematic al
Public ations, Vol . lillY o New Yorka (1949) 1
"Topology
Soci� Collpquium
pp .
Tulane Univer­
.
.. .
.:
Stain, S . K . "On the foundations of quasi- groups . " Tranaat:.ti.ons of the
American Mathemati cal Sooie ty, Vol o 8 5, (19513) , 228-256.