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International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012
454
Intuitionistic Fuzzy Metric Groups
Banu Pazar Varol and Halis Aygün
Abstract1
The aim of this paper is to introduce the structure
of the called “intuitionsitic fuzzy metric group” and
after giving fundamental definitions, to study some
properties of intuitionistic fuzzy metric group.
Keywords: Topology, fuzzy metric spaces, intuitionistic
fuzzy metric spaces, group, normal subgroup.
1. Introduction
Zadeh [18], in 1965, introduced the concept of fuzzy set
as a mapping :
0,1 where is a nonempty set.
Then several researches were conducted on the generalizations of the notion of fuzzy set. The idea of intuitionistic fuzzy set was initiated by Atanassov [2] as a
generalization notion of fuzzy set. By an intuitionistic
fuzzy set we mean an object having the form
,
,
:
where the functions
:
0,1 define respectively, the degree
0,1 and :
of membership and degree of non-membership of the
element
to the set , which is a subset of , and
1. Then, using the
for every
,
concept of intuitionistic fuzzy set several structures have
been constructed. Çoker [6] introduced the concept of
intuitionistic fuzzy topological spaces. The theory of
intuitionistic fuzzy topology was arised from the theory of
fuzzy topology in Chang’s sense [4]. Biswas [3] considered the concept of intuitionistic fuzzy sets with the group
theory. Akram [1] introduced Lie algebras into the intuitionistic fuzzy sets. Then, Chen [5] studied the intuitionistic fuzzy Lie sub-superalgebras. Park [14] defined the notion of intuitionistic fuzzy metric space with
the help of continuous t-norm and continuous t-conorm.
Then, Saadati and Park [16] developed the theory of intuitionistic fuzzy topological spaces (both in metric and
normed).
The idea of fuzzy metric spaces based on the notion of
continuous t-norm was initiated by Kramosil and
Michalek [13]. Then, Grabiec [10] extended the
Corresponding Author: B. Pazar Varol is with the Department of
Mathematics, Kocaeli University, Umuttpe Campus, 41380,
Kocael,Turkey.
E-mail: [email protected], [email protected]
Manuscript received 17 Feb. 2011; revised 19 Jun. 2012; accepted 27
Aug. 2012.
well-known fixed point theorems of Banach and Edelstein to fuzzy metric spaces. The notion of fuzzy metric
space plays a fundamental role in fuzzy topology, so
many authors have studied several notions of fuzzy metric.
Somewhat reformulating the Kramosil and Michalek’ s
definition, George and Veeramani ([8, 9]) introduced and
studied fuzzy metric spaces and topological spaces induced by fuzzy metric. They showed that every metric
induces a fuzzy metric. Then, Gregori and Romaguera [11]
proved that the topology generated by a fuzzy metric is
metrizable. They also showed that if the fuzzy metric
space is complete, the generated topology is completely
metrizable. In 2004, Park [14] introduced the concept of
intuitionistic fuzzy metric spaces as a generalization of
fuzzy metric spaces due to Geroge and Veeramani [8].
Then, Gregori et al. [12] studied on intuitionistic fuzzy
metric spaces and showed that for each intuitionistic
fuzzy metric space, the topology generated by the intuitionistic fuzzy metric coincides with the topology
generated by the fuzzy metric.
Both of fuzzy metric spaces and intuitionistic fuzzy
metric spaces have many applications in various areas of
mathematics. One of them is the concept of fuzzy metric
group. In 2001, Romaguera and Sanchis [15] defined the
concept of fuzzy metric groups which gives us to extensions of classical theorems on metric groups to fuzzy
metric groups. They also studied some properties of the
quotient subgroups of a fuzzy metric group.
In the present paper, we consider the notion of intuitionistic fuzzy metric spaces with the group theory.
The main idea of this work is to extend the fuzzy metric
group to intuitonistic fuzzy metric setting. The structure
of this paper is as follows: In Section 2, as a preliminaries,
we give some definitions and uniform structures of fuzzy
metric spaces and intuitonistic fuzzy metric spaces. In
Section 3, we first introduce the notion of intuitionistic
fuzzy metric group and then study several results of intuitionistic fuzzy metric groups as a generalization of
fuzzy metric groups. In last section, we analyze properties
of the quotient subgroups of an intuitionistic fuzzy metric
group. In this study our basic reference for topology is
Engelking’s book [7].
2. Preliminaries
Throughout this paper let
© 2012 TFSA
be a nonempty set.
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups
Definition 2.1 [17]: A binary operation : 0,1 x 0,1
0,1 is a continuous t-norm if satisfies the following
conditions:
(1)
,
,
0,1
(2)
, , ,
0,1
(3) is continuous
(4)
1
,
0,1
(5)
whenever
and
,
, , ,
0,1 .
Definition 2.2 [17]: A binary operation : 0,1 x 0,1
0,1 is a continuous t-conorm if satisfies the following conditions:
(1)
,
,
0,1
(2)
,
, ,
0,1
(3) is continuous
(4)
0
,
0,1
(5)
whenever
and
,
, , ,
0,1
Note 2.3: The concepts of t-norms (triangular norms) and
t-conorms (triangular conorms) are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively.
Note 2.4: (1) For any ,
0,1 with
, there
0,1 such that
and
exist ,
.
0,1 , there exist ,
0,1 such
(2) For any
and
.
that
Definition 2.5 [8, 9]: A fuzzy metric space is a triple
, , such that is a nonempty set, is a continuous t-norm and
is a fuzzy set on x x 0, ∞ satisfying the following conditions, for all , ,
, ,
0:
(1)
, ,
0;
(2)
, ,
1 if and only if
;
(3)
, ,
, , ;
(4)
, ,
, ,
, ,
;
(5)
, , . : 0, ∞
0,1 is continuous.
If , , is a fuzzy metric space, we say that
,
is a fuzzy metric on .
Example 2.6 [8]: Let ,
be a metric space. Define
be a function on
, ,
0,1 and let
x 0, ∞ defined by
, ,
455
(3)
, ,
1 if and only if
;
(4)
, ,
, , ;
(5)
, ,
, ,
, ,
;
(6)
, , . : 0, ∞
0,1 is continuous;
(7)
, ,
0;
0 if and only if
;
(8)
, ,
(9)
, ,
, , ;
(10)
, ,
, ,
, ,
;
(11)
, , . : 0, ∞
0,1 is continuous.
If , , , , is an intuitionistic fuzzy metric space,
we say that
, , , (or simply
, ) is an intuitionistic fuzzy metric on . The functions
, ,
and
, ,
denote the degree of nearness and the
degree of non-nearness between and with respect to
t, respectively.
Remark 2.8 [12]: If
, , , , is an intuitionistic
fuzzy metric space, then , , is a fuzzy metric space
in the sense of Definition 2.5. Conversely, if , , is
a fuzzy metric space , then
, ,1
, , is an intuitionistic fuzzy metric space, where
1
1
1
for any ,
0,1 .
Remark 2.9: In intuitionistic fuzzy metric spaces X,
, ,.
is non-decreasing and
, ,.
is
.
non-increasing for all ,
Example 2.10 [14] : Let ,
be a metric space. Denote
and
min 1,
for all ,
and
be fuzzy sets on x x 0, ∞
0,1 and let
defined by
, ,
, ,
,
for all , , ,
. Then
, , , , is an intuitionistic fuzzy metric space.
Remark 2.11 [14] : The Example 2.10 holds even with
t-norms
min ,
and the t-conorms
max , . Hence,
,
is an intuitionistic
fuzzy metric with respect to any continuous t-norms and
continuous t-conorms.
Let take
1 in Example 2.10, then we
have
, ,
,
Then
, , is a fuzzy metric space and
, is
called the fuzzy metric induced by .
Definition 2.7 [14]: An intuitionistic fuzzy metric spaces
is a 5-tuple , , , , such that is a nonempty set,
is a continuous t-norm , is a continuous t-conorm and
, are fuzzy sets on x x 0, ∞ satisfying the following conditions, for all , ,
, ,
0:
(1)
, ,
, ,
1;
(2)
, ,
0;
,
,
, ,
,
,
,
This intuitionistic fuzzy metric is called induced intuitionistic fuzzy metric by a metric .
Example 2.12 [14]: Let
. Define
max 0,
1 and
for all
,
0,1 and let
and
be fuzzy sets on
x x 0, ∞ as follows:
International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012
456
,
, ,
,
,
, ,
,
for all ,
and
0 . Then
, , , , is an
intuitionistic fuzzy metric space.
Remark 2.13: In the above example, t-norm
and
t-conorm are not associated. There exist no metric
on satisfying
,
,
, ,
, ,
,
,
where
, , and
, , are as defined in above
example. The above functions
,
is not an intuitionistic fuzzy metric with t-norm and t-conorm defined
as
min ,
and
max , .
Definition 2.14 [14]: Let
, , , , be an intuitionistic fuzzy metric space. Given
,
0,1 and
0. The set
, ,
:
, ,
1
,
, ,
is called the open ball with center
and radius
with
respect to .
Theorem 2.15 [14]: Every open ball
, ,
is an
open set.
The collection
, , :
,
0,1 ,
0 ,
where
, ,
:
, ,
1
,
0, is
, ,
for all
,
0,1 and
a base of a topology on
and this topology is denoted
. Hence, each intuitionistic fuzzy metric
by
,
on .
,
on
generates a topology
,
Notice that the collection
, , :
,
is also a base for
, .
The topology
is Hausdorff and first countable.
,
If
,
is a metric space, then the topology induced
induced by
by
coincides with the topology
,
,
.
the intuitionistic fuzzy metric
:
, where
The family
,
x
, ,
,
1
, ,
is a base for uniformity
on
,
whose induced topology coincides with the topology
and hence
,
is metrizable.
,
,
Theorem 2.16 [14]: Every intuitionistic fuzzy metric
spaces is Hausdorff.
Remark 2.17 [14]: Let
,
be a metric space. Let
, ,
,
,
, ,
,
,
,
be the intuitionistic fuzzy metric defined on . Then the
topology
induced by the metric
and the topology
induced by the intuitionistic fuzzy metric
,
,
are the same.
Definition 2.18 [14]: Let
, , , , be an intuitionistic fuzzy metric space.
on is called Cauchy sequence if
(1) A sequence
such that
for each
0 and
0 there exist
, ,
1
and
, ,
for all
.
,
(2)
, , , , is called complete intuitionistic fuzzy
metric space if every Cauchy sequence is convergent with
respect to
, .
If
, , , ,
is a complete intuitionistic fuzzy
metric space we say that
,
is a complete intuitionistic fuzzy metric on .
If
is a Cauchy sequence on
,
, then
,
is a Cauchy sequence on the intuitionistic fuzzy
metric space
, , , , , so the uniform space
is
complete
whenever
, , , ,
is a
,
,
complete.
Definition 2.19 [15]: A fuzzy metric group is a 4-tuple
,·, , such that
, , is a fuzzy metric space
is a topological group.
and
,·,
3. Intuitionistic Fuzzy Metric Groups
Definition 3.1: A topological space
, is called intuitionistic fuzzy metrizable if there exist an intuitionistic
fuzzy metric
,
on such that the topology
,
induced by
,
coincides with . The intuitionistic
fuzzy metric is called compatible with .
A topological group
is called intuitionistic fuzzy
metrizable if it is intuitionistic fuzzy metrizable as a
topological spaces.
Definition 3.2: An intuitionistic fuzzy metric group is a
6-tuple
,·, , , , such that
, , , , is an inis a
tuitionistic fuzzy metric space and
,·,
,
topological group.
Definition 3.3: An intuitionistic fuzzy metric
,
on
a group
is called left invariant (right invariant) if
, ,
, , and
, ,
, ,
(
, ,
, , and
, ,
, , )
whenever , ,
and
0.
Theorem 3.4: If
is an intuitionistic metrizable topological group and
,
is a compatible left (right)
invariant intuitonistic fuzzy metric on , then the uniinduced by
,
coincides
form structure ,
,
with the left (right) uniform structure , .
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups
457
Proof: We must show that the two uniformities have a
common base. Let, for each
, take a basic
neighborhood of
, ,
,
:
, ,
1
:
.
, ,
Hence, for
, :
, :
:
,
, ,
,
1
, ,
is left invariant,
,
Since
1
,
,
,
,
,
,
.
In the following, we use next well-known theorem.
Theorem 3.5: Let
,· be a group with identity
and
let
be a family of subsets of
which satisfies the
following properties:
(1)
(2) If ,
, then there exists
such that
(3) for every
, there exist
such that
(4) for every
and every
, there exist
such that
.
Then, there exist a unique Hausdorff topological group
on which has
as a local base of .
Lemma 3.6: Let ,· be an group and
,
be a left
invariant intuitionistic fuzzy metric on
. Then
and
, , is symmetric for all
, ,
, ,
.
, ,
Proof: Let
1
, ,
, ,
,
for all
. Then
, ,
and
1
,
.
and
1
, ,
,
and
,
, , .
From the following equalities we see that the second part
of the lemma.
1
1 1
:
, ,
, ,
1
1
:
1
and
,
1
and
, ,
,
1
1
1
,
,
1
1
1 1
, ,
:
1 1
, ,
.
1 1
, ,
Proposition 3.7: A topological group approves a compatible left invariant intuitionistic fuzzy metric if and only
if it approves a compatible right invariant intuitionistic
fuzzy metric.
Proof: Let
,
be a compatible left invariant intuitionistic fuzzy metric on a topological group . Define
two functions
and
on x x 0, ∞ as follows:
,
,
, ,
,
,
x, y, t
whenever ,
and
0.
,
is a right invariant intuitionistic fuzzy metric on
. Indeed,
,
,
, ,
,
,
,
,
, ,
Similarly,
, ,
x, y, t .
Now, we will show that it is a compatible one.
Since
, ,
is symmetric for all
, then we
, ,
have
Hence,
, ,
, ,
.
is a base of neighbourhoods
of
and that every topological group is homogeneous.
From now on the paper all topologies are assumed to
be .
Theorem 3.8: Let ,· be a group and
,
be a left
and right invariant intuitionistic fuzzy metric on . Then
,·, , , , is an intuitionistic fuzzy metric group.
Proof:
Let
, , :
be
the
neighbourhood base of for the intuitionistic fuzzy topology
, .
We need to show that
satisfies conditions (1)-(4) in the
is a Hausdorff topology and
Theorem 3.5. Since
,
is a neighbourhood base of , we only check the conditions (3)-(4).
(3) Let
, ,
. From the Lemma 3.6,
, ,
is symmetric. So, if we find
such that
, ,
.
, ,
, , , then it is sufficient.
such that for each
,
Let
,
and let
1
1
1
max
,
, 2 . Then, for each
, , ,
International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012
458
, . ,
, . ,
, ,
, ,
, . ,
1
, ,
1
1
1
1
and
, . ,
, . ,
, ,
, ,
, ,
So, .
(4) Let
.
Then,
,
and
.
, ,
.
, ,
, ,
1
.
, ,
, ,
, . ,
1
, ,
,
,
, ,
,
,
1
. Hence,
and
.
So,
, , .
This completes the proof.
4. Intuitionistic Fuzzy Metric On Quotient
Groups
Let
be a complete topological group and
be a
closed normal subgroup of . In general, the qutient
group
is not complete. But if
is metrizable , the
quotient group is complete.
Moreover, if
and
are complete, then the group
is complete. [15]
Let ,
be a metric group and
be a closed normal subgroup of . Then the metric induced by on
the quotient group
is defined by
,
inf
, :
,
.
induced by
on
is deThe fuzzy metric
fined by following:
, ,
sup
, , :
,
,
0
The induced fuzzy metric
, on
for a closed
normal subgroup
of a fuzzy metric group ,·, , is
defined by
, ,
sup
, , :
,
,
0. [15]
Now we consider these results for intuitionistic fuzzy
metric groups.
The intuitionistic fuzzy metric
,
induced by
on
is defined by the following:
, ,
sup
, , :
,
, ,
inf
, , :
,
,
0.
Let be an intuitionistic fuzzy metrizable topological
group and
,
be a left invariant compatible intuitionistic fuzzy metric on . Let
be a closed normal
subgroup of . The induced intuitionistic fuzzy metric
,
on
x
x 0, ∞ is defined by
, ,
sup
, , :
,
, ,
inf
, , :
,
,
0
Theorem 4.1:
,
satisfies the conditions (2)-(5) and
(7)-(10) which is given in the definition of intuitonistic
fuzzy metric.
Proof: Conditions (2), (4), (7) and (9) are obvious.
1 and
, ,
If
and
0, then
, ,
0 for all
0.
Conversely, let cosets ,
,
, ,
1 and
, ,
0 for all
0. Then there exist two se,
satisfying
quences
, ,
and
, ,
for all
1
.
We will show that
converges to .
Let take a neighbourhood
, ,
of the identity .
Consider
. Then,
,
,
, ,
,
,
1
1
and
,
,
,
,
,
,
Therefore, for each
,
, ,
and
converges to . Since
thus, the sequence
is normal
.
for each
. Since
is
So,
closed, then
. Thus,
.
Similarly, we can see that the sequence
converges to and
.
Thus (3) and (8) are provided.
Let ,
. Since
and are continuous, we
have
, ,
, ,
, ,
Banu Pazar Varol and Halis Aygün: Intuitionistic Fuzzy Metric Groups
459
, ,
and
, ,
, ,
, ,
, ,
for each
,
. Thus (5) and (9) are provided.
In general,
and
do not satisfy conditions (6) and
(11) of the definition of an intuitionistic fuzzy metric.
Example 4.2: Let ( ,+) be the usual group of real numbers and let consider the continuous t-norm and the continuous t-conorm on the unit interval I=[0,1] defined by
and
min 1,
.
and
be two sequences of conLet
tinuous non-decreasing and non-increasing functions
from 0, ∞ into
,
satisfying that the functions
and
are not continuous, respectively.
Let
and
be two functions from
0, ∞ into [0,1] defined by
, ,
,
max
,
1
1,
, ,
,
min
,
0,
where [x] means for the integer part of the real numbers x.
Claim 1.
,
is an intuitionistic fuzzy metric on
0, ∞ such that
0,
,
, ,
and
0,
,
, ,
for each
, ,
0, ∞ .
Proof: We only need to check the conditions
, ,
, ,
, ,
and
, ,
, ,
, ,
where , ,
and ,
0.
Case 1. If
, ,
, ,
1 and
, ,
, ,
0 , then
and consequently
, ,
1 and
, ,
0.
and
Case 2. Let
, ,
1,
, ,
for some
.
, ,
0,
, ,
Since
, then
,
,
.
Beis non-decreasing and
is non-increasing
cause of
the result is provided.
and
Case 3. If
, ,
1,
, ,
for some
, then
, ,
0,
, ,
the result is similarly to the Case 2.
,
, ,
and
Case 4. If
, ,
,
, ,
for
some
, ,
,
, then
for every
0 and
.
and
Let
for every
0
and
be two functions
x 0, ∞ into [0,1] defined by
, ,
sup
, , :
,
, ,
inf
, , :
,
from
.
x
,
Claim 2. The functions
,· and
, ,· are not continuous on 0, ∞ , where
notes the set of integer numbers.
Proof: Let
,
and
and
,
, then
:
,
Let
de-
,
, then
:
and
, ,
are not
Consequently,
, ,
continuous.
From Theorem 4.1 and Example 4.2, we see that definition of intuitionistic fuzzy metric space must be modified by replacing conditions (6) and (11) by
6 the function
, ,· is lower semicontinuous for
each ,
, ,· is upper semicontinuous for
11 the function
each ,
By a lower-upper intuitionistic fuzzy metric space we
mean a 5-tuble , , , , such that is a nonempty
set, is a continuous t-norm, is a continuous t-conorm,
and
are two fuzzy set on x x 0, ∞ satisfying
conditions (1)-(5), (7)-(10) and 6 , 11 above. Then,
,
is called a lower-upper intuitionistic fuzzy metric
on .
In [14] Park proved that if
,
is an intuitionistic
fuzzy metric, then
, ,· is non-decreasing and
, ,· is non-increasing for all ,
. Since for any
lower-upper intuitionistic fuzzy metric
,
and any
,
0 with
, we have
, ,
, ,
, ,
, ,
and
, ,
, ,
, ,
, ,
for all ,
, we obtain that Park’s result remains valid
for lower-upper intuitionistic fuzzy metric.
Let see the following results:
(1) Every lower-upper intuitionistic fuzzy metric
,
on
such
on
induces a metrizable topology
,
that
, , :
for each
.
is a neighbourhood base of
International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012
460
(2) If
,
is a lower-upper intuitionistic fuzzy metric
:
is a base for a uniformity on
on , then
compatible with
, where
,
,
:
, ,
,
1
, ,
for all
.
If we introduce the notions of a lower-upper intuitionistic fuzzy metric group and of a lower-upper intuitionistic fuzzy metrizable topological group, it is easily seen
that the results which we have obtained above also hold
for lower-upper intuitionistic fuzzy metric groups and for
lower-upper intuitionistic fuzzy metrizable topological
groups, respectively.
Let
,·, , , ,
be a lower-upper intuitionistic
fuzzy metric group and
be a closed normal subgroup
of . Then we define the set
, ,
by
1
1
1 1
,
, ,
1
, ,
, ,
1
1
Lemma 4.3: Let
,·, , , , be a lower-upper intuitionistic fuzzy metric group such that
,
is left
invariant and let
be a closed normal subgroup of .
Then
, ,
, ,
Proof: Let
for each
, ,
. If
, ,
, ,
1
, ,
, ,
.
, ,
So,
,
,
,
neighbourhoods of
, we can find
,
,
,
.
1
,
,
, ,
in
is a base of
for the quotient topology.
By Lemma 4.3 every element of
longs to the family
, ,
. By Lemma 4.3,
is a neighbourhood of
for the in-
tuitionistic fuzzy topology. This completes the proof.
5. Conclusions
Intuitionistic fuzzy sets have the forceful ability of
explaining uncertainties compared with fuzzy sets.
In this paper, we generalize the approach introduced
by Romaguera et al. [15]. In this manner, we define
the notion of intuitionistic fuzzy metric group and
study some properties.
Acknowledgment
The authors are highly grateful to the referees for
their valuable suggestions.
References
, we obtain that
.
and
Theorem 4.4: Let
,·, , , , be a lower-upper intuitionistic fuzzy metric group such that
,
is left
invariant and
is a closed normal subgroup of . Then
the topology of the lower-upper intuitionistic fuzzy metric group
,·, , , , coincides with the quotient
topology.
Proof: The family
. Then,
, ,
[1]
,
,
, ,
and
, ,
,
such that
, then
.
Conversely, if
and
such that
, ,
and
1
Since
.
prove that every neighbourhood
of
for the quotient
topology is also a neighbourhood of
for the intuitionistic fuzzy topology. Let
be a set such that
is a neighbourhood of in . We can choose
, ,
, ,
, ,
be-
. So, we must
M. Akram, “Intuitionistic fuzzy Lie ideals of Lie
algebras,” Int. Journal of Fuzzy Math., vol. 16, no.
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