A study about
β-fuzzy topological group
Assist. Prof. Dr. Munir Abdul Khalik Alkhafaji
Al-Mustansiriyah University
College of Education
Department of Math.
E-mail:[email protected]
Abstract
In this paper we considered fuzzy topological group and  -fuzzy
topological group, and we proved some results about the connection
between these two concepts.
Keywords: fuzzy topological group, Quotient fuzzy topological group,
 -fuzzy topological group
1. Introduction
In his classical paper [8] in 1965 Zadeh introduced the notation of
fuzzy sets and fuzzy set operation. Subsequently, Chang [4], applied some
basic concepts from general topology to fuzzy sets, he also developed a
theory of fuzzy topological spaces. In this paper we study some properties
of fuzzy topological groups, and by depending on concepts introduced in
1
[7] and [9]; we define the new concept named by  -fuzzy topological
group, also we proved some properties on it.
2. Preliminary
Definition 2.1. [5]
Let 𝑋 be a set and
the unit interval [0.1].a fuzzy set 𝐴̃ in 𝑋 is
characterized by a membership function 𝑀𝐴̃ which is associate with each
point 𝑥 ∈ 𝑋 its “grade of membership” 𝑀𝐴̃ ∈ Ι
Definition 2.2. [12]
~
Let f be a mapping from X to a set Y. Let B be a fuzzy set in Y,
~
with membership function M B~ . Then the inverse image of B written
~
f 1 ( B ) is the fuzzy set in X with membership function defined by
Mf
1
~
(B)
( x)  M B~ ( f ( x))
for all x  X
~
Conversely, let A be a fuzzy set in X with membership function M A~ then
the image of
~
~
A written f ( A) , is the fuzzy set in Y with membership
function defined by
sup M A~ ( x) : x  X , f ( x)  y if f 1 ( y )  
M f ( A~ ) ( y )  
0
otherwise
for all y  Y where f 1 ( y)  x /
f ( x)  y
Theorem 2.3. [12]
Let f be a function from X to Y and I be any index set. Then the
following Statements are true.
2
~
~
~
1-if A  X then f ( A) C  f ( A C )
~
~
~
2-if B  Y then f 1 ( B C )  f 1 ( B ) C
~ ~
3-if A1 , A2  X and
~ ~
4-if B1 , B2  Y and
~
~
~
~
A1  A2 then f ( A1 )  f ( A2 )
~
~
~
~
B1  B2 then f 1 ( B1 )  f 1 ( B2 )
~
~
~
5-if A  X then A  f 1 ( f ( A))
~
~
~
6- if B  X then f ( f 1 ( B ))  B
~
~
~
7-if Ai  X for every i  I then f ( Ai )   f ( Ai )
iI
iI
~
~
~
8-if Bi  Y for every i  I then f 1 ( Bi )   f 1 ( Bi )
iI
iI
~
~
~
9- if Bi  Y for every i  I then f 1 ( Bi )   f 1 ( Bi )
iI
iI
~ ~
~ ~
~
~
10- if A1 , A2  X then f ( A1  A2 )  f ( A2 )  f ( A1 )
~
~
~
11- if f is one to one and A  X then A  f 1 ( f ( A))
~
~
~
12-if f is onto and B  X then f ( f 1 ( B ))  B
~
13-Let g be a function from Y to Z if B  Z then
~
~
~
~
~
( gof ) 1 ( B )  f 1 ( g 1 ( B )) and if A  X then ( gof )( A)  g ( f ( A))
~
~
~
14- if f is bisection then for A  X then f ( A) C  f ( A C )
Definition 2.4.[11]
~
A fuzzy topology is a family T of fuzzy sets in X which satisfies the
following conditions
~
1- kc  T c  [0,1]
~ ~ ~
~ ~ ~
2- if A, B  T then A  B T
~ ~
~ ~
3-if Ai  T i  I then  Ai  T
iI
3
~
~
the pair ( X ,T ) is a fuzzy topological space (FTS).Every member of T is
~
~
called a T -open fuzzy set in ( X ,T ) (or simply open fuzzy set) and
complement of a open fuzzy set is called closed fuzzy set
Definition 2.5. [9]
A fuzzy topology 𝑇̃ on a group 𝐺 is said to be compatible if the
mapping
𝑔: (𝐺 × 𝐺, 𝑇̃ × 𝑇̃) → (𝐺, 𝑇̃) , 𝑔(𝑥, 𝑦) = 𝑥𝑦
and
ℎ: (𝐺, 𝑇̃) → (𝐺, 𝑇̃), ℎ(𝑥) = 𝑥 −1
are fuzzy continuous
Definition 2.6. [9]
A group 𝐺 equipped with a compatible fuzzy topology 𝑇̃ on 𝐺 is
called a fuzzy topological group (FTG)
Quotient group2.7
Let 𝐺 be a group and 𝐻be a normal subgroup of 𝐺 for each 𝑥 ∈ 𝐺, Let 𝑥𝐻
denote the unique coset to which 𝑥 belongs in the decomposition of 𝐺 into
pair wise disjoint cosets. Let 𝜑: 𝐺 → 𝐺 ⁄𝐻 define by 𝜑(𝑥) = 𝑥𝐻
It is easy to check 𝜑 is homomorphism of 𝐺 into 𝐺 ⁄𝐻.
To show this let 𝑥, 𝑦 ∈ 𝐺 , 𝜑(𝑥𝑦) = 𝑥𝑦𝐻 = 𝑥𝐻𝑦𝐻 = 𝜑(𝑥)𝜑(𝑦) Hence 𝜑
is homomorphism.
4
̃ as follows:2.8
Quotient fuzzy topology on 𝐺 ⁄𝐻
̃ in 𝐺 ⁄𝐻
̃ is open fuzzy if and only if 𝜑 −1 (𝑊
̃ ) is open
A fuzzy set 𝑊
̃ ; 𝑎 ∈ 𝐴̃}is open fuzzy if and only if ∪
fuzzy in 𝐺.Note that 𝐴̃̇ = {𝑎𝐻
̃ ; 𝑎 ∈ 𝐴̃} is an open fuzzy in 𝐺
{𝑎𝐻
Theorem 2.9. [9]
Let (𝐺, 𝑇̃) be an FTG and a, b∈ 𝐺 then
1- The translation maps
𝑟𝑎 : (𝐺, 𝑇̃) → (𝐺, 𝑇̃ ) , 𝑟𝑎 (𝑥) = 𝑥𝑎
and
𝑙𝑎 : (𝐺, 𝑇̃) → (𝐺, 𝑇̃ ) , 𝑙𝑎 (𝑥) = 𝑎𝑥
2- The inversion map
𝑓: (𝐺, 𝑇̃) → (𝐺, 𝑇̃ ) , 𝑓(𝑥) = 𝑥 −1
and the map
𝜑: (𝐺, 𝑇̃) → (𝐺, 𝑇̃) , 𝜑(𝑥) = 𝑎𝑥𝑏
are all fuzzy homeomorphisms.
3. Fuzzy topological group
Theorem 3.1.
Let 𝐺 be a group having fuzzy topology 𝑇̃ then (𝐺, 𝑇̃) is FTG if and
only if the map 𝑔: (𝐺 × 𝐺, 𝑇̃ × 𝑇̃) → (𝐺, 𝑇̃) define by 𝑔(𝑥, 𝑦) = 𝑥𝑦 −1 is
fuzzy continuous
5
Theorem 3.2.
Let (𝐺, 𝑇̃) be FTG, 𝐴̃, 𝐵̃  𝐺, 𝑔 ∈ 𝐺 then
1- 𝐴̃ open fuzzy implies 𝐴̃𝑔 , 𝑔𝐴̃ , 𝑔𝐴̃𝑔−1 𝑎𝑛𝑑 𝐴̃−1 are open fuzzy
2- 𝐴̃ closed fuzzy implies 𝐴̃𝑔 , 𝑔𝐴̃ , 𝑔𝐴̃𝑔−1 𝑎𝑛𝑑 𝐴̃−1 are closed fuzzy
3- 𝐴̃ open fuzzy implies 𝐴̃𝐵̃𝑎𝑛𝑑𝐵̃𝐴̃ open fuzzy
4- 𝐴̃ closed fuzzy and 𝐵̃ finite implies 𝐴̃𝐵̃𝑎𝑛𝑑𝐵̃𝐴̃ closed fuzzy .
Proof:
1 and 2
𝑟𝑎 , 𝑙𝑎 𝑓, 𝜑
being fuzzy homeomorphism. Are all open fuzzy
and closed fuzzy
3-𝐴̃𝐵̃ =∪ {𝐴̃𝑏; 𝑏 ∈ 𝐵̃} is the union of open fuzzy sets and hence open
fuzzy similarly for 𝐵̃𝐴̃
4-Similarly as above. ∎
Definition 3.3.
A FTG 𝐺 is called fuzzy homogenous if for any 𝑎, 𝑏 ∈ 𝐺 there is a fuzzy
homeomorphism 𝑓: 𝐺 → 𝐺 𝑠. 𝑡 𝑓(𝑎) = 𝑏.
Theorem 3.4
A FTG is a fuzzy homogenous space.
Proof:
Let 𝐺 be a FTG and 𝑥1 , 𝑥2 ∈ 𝐺 𝑡𝑎𝑘𝑒 𝑎 = 𝑥1−1 𝑥2
𝑓(𝑥) = 𝑟𝑎 (𝑥) = 𝑥𝑎 = 𝑥𝑥1−1 𝑥2
𝑖𝑚𝑝𝑙𝑖𝑠𝑒 𝑓(𝑥1 ) = 𝑥2 by theorem 2.9 𝑓
is fuzzy homeomorphism ∎
Theorem 3.5.
A non trivial FTG has no fixed point properties
6
Proof:
Let 𝐺 be a FTG, 𝑎 ∈ 𝐺 with 𝑎 ≠ 𝑒 now the map 𝑟𝑎 : 𝐺 → 𝐺 is fuzzy
continuous suppose 𝑟𝑎 (𝑥) = 𝑥 for some 𝑥 ∈ 𝐺 𝑥𝑎 = 𝑥 Implies
𝑎=𝑒
which is contradiction to the concepts that 𝑟𝑎 has no fixed point, hence 𝐺
has no fixed point properties ∎
Theorem 3.6.
Every open fuzzy subgroup of FTG is closed fuzzy
Proof:
̃ be open fuzzy subgroup of 𝐺
Let 𝐺 be a FTG and 𝐻
̃ =∪ {𝑔𝐻
̃ ; 𝑔 H
̃ }={𝑟𝑔 (𝑥); 𝑔 H
̃ } which is open fuzzy by theorem 3.2
𝐺−𝐻
̃ is closed fuzzy ∎
There fore 𝐻
Theorem 3.7.
̃ be a fuzzy subgroup of 𝐺.let 𝐺 ⁄𝐻
̃ be quotient
Let 𝐺 be FTG and 𝐻
space, endowed with the quotient topology and 𝜑 the canonical mapping of
̃
𝐺 into 𝐺 ⁄𝐻
Then
1- 𝜑 is onto
2- 𝜑 is fuzzy continuous
3- 𝜑 is fuzzy open
Proof:
̃ implies 𝑝 = 𝑥𝐻
̃ = 𝜑(𝑥) we get 𝜑 is onto
1- Let 𝑝 ∈ 𝐺 ⁄𝐻
2- It is clear from definition
3- Let 𝐴̃ be any open fuzzy in 𝐺 we have show that 𝜑(𝐴̃) is open fuzzy in
̃ i.e. 𝜑 −1 (𝜑(𝐴̃)) is open fuzzy in 𝐺.
𝐺 ⁄𝐻
̃ 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑎 ∈ 𝐴̃} = 𝐴̃𝐻
̃
𝜑 −1 (𝜑(𝐴̃)) = {𝑥; 𝑥 ∈ 𝑎𝐻
7
̃ 𝐴̃ is open fuzzy, therefore 𝑓 is fuzzy open mapping ∎
By theorem 3.2 𝐻
Theorem 3.8.
̃ be fuzzy subgroup of 𝐺 then 𝐺 ⁄𝐻
̃
Let 𝐺 be FTG and 𝐻
is fuzzy
̃ is open fuzzy
discrete topological space if and only if 𝐻
Proof:
̃ be a fuzzy discrete topological space then each subset of
Let 𝐺 ⁄𝐻
̃ is open fuzzy and hence each singleton is open fuzzy in particular 𝐻
̃
𝐺 ⁄𝐻
̃ , there for 𝐻
̃ is open fuzzy
is fuzzy point of 𝐺 ⁄𝐻
̃ be open fuzzy so is 𝑥𝐻
̃ open fuzzy for each 𝑥 ∈ 𝐺 by
Conversely, let 𝐻
̃
theorem 3.2 This show that each singleton {𝑥̇ } is open fuzzy in 𝐺 ⁄𝐻
,
̃ is fuzzy discrete space ∎
Therefore 𝐺 ⁄𝐻
Theorem 3.9
̃ normal fuzzy subgroup of 𝐺 and 𝐺 ⁄𝐻
̃
Let 𝐺 be FTG and 𝐻
endowed with the quotient fuzzy topology, is FTG.
Proof:
̃ × 𝐺 ⁄𝐻
̃ 𝑜𝑛𝑡𝑜 𝐺 ⁄𝐻
̃
Let 𝐺 be FTG to prove (𝑥̇ , 𝑦̇ ) → 𝑥̇ 𝑦̇ −1 of 𝐺 ⁄𝐻
̃̇ be any open fuzzy of
is fuzzy continuous. Let 𝑊
𝑥̇ 𝑦̇ −1 where 𝑥̇ =
̃̇ ) is open fuzzy in 𝐺 and 𝑥𝑦 −1 ∈ 𝜑 −1 (𝑊
̃̇ ) ,
̃ 𝑎𝑛𝑑 𝑦̇ = 𝑦𝐻
̃ clearly 𝜑 −1 (𝑊
𝑥𝐻
̃, 𝑉̃ 𝑖𝑛 𝐺 such that 𝑥 ∈ 𝑈
̃, 𝑦 ∈ 𝑉̃ and
since 𝐺 is FTG ∃ 𝑈
𝑥𝑦 −1 ∈
̃̇ ) 𝑥̇ 𝑦̇ −1 ∈ 𝜑(𝑈
̃̇ )) = 𝑊
̃̇ by theorem
̃𝑉̃ −1  𝜑 −1 (𝑊
̃)𝜑−1 (𝑉̃ ) ⊆ 𝜑 (𝜑 −1 (𝑊
𝑈
̃) and
3.7 𝜑is open fuzzy mapping we get there exist an open sets 𝜑(𝑈
̃ is FTG ∎
𝜑(𝑉̃ −1 ) of 𝑥̇ 𝑎𝑛𝑑 𝑦̇ −1 respectively there fore 𝐺 ⁄𝐻
8
4.  - fuzzy topological group
Definition 4.1
~
~
Let G be a group and (G, T ) be a fuzzy topological space. (G, T ) is
called  - fuzzy topological group or  -FTG for short if
~
~
~
the maps g : (G,T )  (G,T )  (G,T ) defined by g ( x, y)  xy and
~
~
h : (G,T )  (G,T ) defined by h( x)  x 1 are fuzzy M -continuous
Theorem 4.2.
~
Let G be group having fuzzy topology. (G, T ) is  -FTG if and only
~
~
~
if the mapping g : (G,T )  (G,T )  (G,T ) is defined by g ( x, y)  xy 1 is
fuzzy M -continuous.
Theorem 4.3
~
Let a be a fixed element of  -fuzzy topological group (G, T ) then the
~
mappings ra ( x)  xa, la ( x)  ax, f ( x)  x 1 and g ( x)  axa1 of (G, T ) onto
~
(G, T ) are fuzzy M -homeomorphisms of G
Proof:
It is clear ra is one to one and onto
~
~
Let W be any  -open fuzzy set containing xa since (G, T ) is  -FTG there
~
~
~
~
~
exist  -open fuzzy set U of x s.t Ua  W we get ra (U )  W
9
Hence ra is fuzzy M -continuous, It is easy to see that ra
1
of ra is the
mapping ra ( x)  xa 1 which is fuzzy M -continuous by the same way of
1
ra ,Therefore ra is M -homeomorphism .Similarly for l a
Finally the composition of two fuzzy M -homeomorphisms
r 1a ( x)  xa 1 and la ( x)  ax is fuzzy M -homeomorphisms
Hence g ( x)  ra  la ( x)  axa 1 is fuzzy M -homeomorphisms ∎
1
Corollary 4.4
~
~
~
Let F be  -closed fuzzy, P be an  -open fuzzy and A any
~
subset of  -FTG (G, T ) and a  G then
~ ~
~
1- Fa, aF and F 1 are  -closed fuzzy
~
~ ~ ~ ~~
~
2- Pa , aP , A P , PA and P 1 are  -open fuzzy
Corollary 4.5
~
Let (G, T ) be  -FTG for any x1 , x2  G there exists a fuzzy M -
homeomorphism 𝑓 𝑜𝑓 𝐺 𝑠. 𝑡 𝑓(𝑥1 ) = 𝑥2
Proof:
~
~
1
Let x1 x2  a  G and consider the mapping f : (G,T )  (G,T ) defined
by f ( x)  xa then f is fuzzy M -homeomorphism by theorem 4.3
f ( x1 )  x2
∎
A space for which corollary 4.5 is true is called a fuzzy𝛽-homogeneous
space
Theorem 4.6.
~ ~
~
Let (G, T ) be  -FTG and A, H are fuzzy subset of G then
10
~
~
1- cl (aA a 1 )  acl ( A)a 1 where a  G is definite point
~
~ ~
~
~~
~
~
2- if cl ( A)  cl ( H )  cl ( A  H ), cl ( A) cl ( H )  cl ( AH ) and
~
~~
~
cl ( A) cl ( H 1 )  cl ( A H 1 )
Proof:
~
By corollary 4.4 a cl ( A)a 1 is  -closed fuzzy set, since is the
~
~
~
smallest  -closed fuzzy set containing aA a 1 , cl (aA a 1 )  acl ( A)a 1
~
~
Let f : (G,T )  (G,T ) be a map defined by f ( x)  axa1
by theorem 4.3 f is fuzzy M -homeomorphism by lemma2.5in paper [7]
~
~
f ( cl ( A))  cl ( f ( A)) thus
~
~
~
~
acl ( A)a 1  cl (aA a 1 ) we get acl ( A)a 1  cl (aAa 1 )
~
~
~
For proof part 2 by theorem 4.6 the map g : (G,T )  (G,T )  (G,T ) is
defined by g ( x, y)  xy 1 is fuzzy M -continuous, since
~
~
~
~
~
~
~
~
cl ( A)  cl ( H )  cl ( A  H ) , f ( cl ( A), cl ( H ))  f ( cl ( A  H ))
~ ~
~ ~
Since f is fuzzy M -continuous f ( cl ( A  H ))  clf ( A, H )) By lemma
~
~~
~
2.5 in paper [7] thus cl ( A) cl ( H ) 1  cl ( A H 1 ) ,for x G
~
~
M cl ( H~ ) ( x)  M K~ :H~  K~ , K~ is  closed ( x)  inf M k~ ( x) : H 1  K i
1


1
i
i
i

~ ~ 1
= inf M K~ ( x 1 ) : H  K i = M K~
1
i
~ 1
1 ~
i :H  K i
I

1
1
( x )  M cl ( H~ ) ( x )  M cl ( H~ ) ( x)
1
~
~~
~
~
~
We get cl ( H 1 )  cl ( H ) 1 hence cl ( A) cl ( H 1 )  cl ( A H 1 )
~
~~
~
Similarly for cl ( A) cl ( H )  cl ( A H )
∎
Theorem 4.7
~
~ ~
~
~
~
Let (G, T ) be an  -FTG and cl ( A)  cl ( H )  cl ( A  H ) if H is
~
~
a fuzzy subgroup of G then cl (H ) is also fuzzy subgroup of G , if H is a
11
~
fuzzy normal subgroup of G then cl (H ) is also fuzzy normal subgroup of
G
Proof:
~~ ~
~~
~
In [10] HH  H  cl ( HH )  cl ( H ) by above theorem
~
~
~~
~
~
~
cl ( H )cl ( H )  cl ( HH ) we get cl ( H )cl ( H )  cl ( H ) …(1)
~
Since H is fuzzy subgroup M H~ ( x)  M H~ ( x 1 )  M H~ ( x) for all x G
1
~ ~
~
~
H  H 1 and hence cl ( H )  cl ( H 1 ) we may show that for every x  G
M cl ( H~ ) ( x)  M cl ( H~ ) ( x) using the same method a above
1
1
M cl ( H~ ) ( x)  M cl ( H~ ) ( x)  M cl ( H~ ) ( x 1 ) …(2) from (1)&(2) and in [10]
1
~
~
cl (H ) is fuzzy subgroup of G ,Let H be a fuzzy normal subgroup of G
then M H~ (ab)  M H~ (ba) for any a, b  G and hence
M xH~x ( z )  M xH~ ( zx )  M H~ ( x 1 zx)  M H~ ( x 1 xz )  M H~ ( z )
1
~
~
~
~
Hence xHx 1  H we get cl ( xHx 1 )  cl ( H ) by theorem 4.6 hence
~
~
xcl ( H ) x 1  cl ( H ) for every x  G
M cl ( H~ ) ( xy)  M xcl ( H~ ) x ( xy)  M cl ( H~ ) x ( x 1 xy)  M cl ( H~ ) x ( y)  M cl ( H~ ) ( yx)
1
1
1
~
We get cl (H ) is fuzzy normal subgroup of G ∎
Theorem 4.8.
~
~
Every  -open fuzzy subgroup H of  -FTG (G, T ) is  -closed
fuzzy
Proof:
~
for each x  G, xH is  -open fuzzy by corollary (4.4) hence
~
~
~
H  ( xH ) C is  -closed fuzzy because  xH is  -open fuzzy , where the
~
union is taken over all pair wise fuzzy cosets different from H ∎
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