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Generating fuzzy factor groups and the fundamental theorem of isomorphism
IRRIV
sets and systems ELSEVIER
Fuzzy Sets and Systems 82 (1996) 357-360
Generating fuzzy factor groups and the fundamental theorem of isomorphism De-Gang Chen a'*, Wen-Xiang Gu b =Department of mathematics, JinZhou Teacher's College, .linZhou, LiaoNing, 121003, Peoples' Republic of China bDepartment of computer science, Northeast Normal University, ChangChun, JiLin, 130024, Peoples' Republic of China Received July 1994; revised June 1995
Abstract In this paper we study how a fuzzy set can generate a fuzzy factor group, we also study the product structure of the generating fuzzy factor group. At the end of this paper we prove the first fundamental theorem of isomorphisms of fuzzy groups.
Keywords: Fuzzy subgroup; Generating fuzzy group; Fuzzy factor group; Fundamental theorem of homomorphisms
1. Introduction
Since Rosenfeld gave the concept of fuzzy subgroup in his pioneering paper [4] in 1971, many mathematicians around the world have since studied deeply the properties of fuzzy groups. Research in fuzzy groups have produced a large number of papers. The references [1-6] represent some of the more recent works. In [1, 2] the concept of fuzzy factor group is given and the fundamental theorem of homomorphisms of fuzzy groups is proven, the product structure of fuzzy factor groups is also discussed. For the first time Zhu Nan-De introduced the concept of generating fuzzy subgroup and studied how a general fuzzy set could generate a fuzzy group. Since we have the definition of fuzzy
*Corresponding author.
factor group, we study the properties when a fuzzy set generate a fuzzy factor group. By use of the fundamental theorem of homomorphisms of fuzzy groups which has been proved in [1], we prove the first fundamental theorem of isomorphisms of fuzzy groups.
2. Preliminaries
Definition 2.1 (Rosenfeld [4]). Let A, B be fuzzy sets of G, if for any x, y E G, (1) A(xy)/> min{A(x), A(y)};
(2) a(x-1)/> a(x), then A is called a fuzzy subgroup of G. Definition 2.2 (Zadeh [7]). Let A, B be fuzzy sets of G, if for any x E G, A(x) .~ B(x), then we call A is contained by B, denoted as A _ B.
0165-0114/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 5 ) 0 0 2 7 1 - 5
358
D.-G. Chen, W.-X. Gu / Fuzzy Sets and Systems 82 (1996) 3 5 7 - 3 6 0
Definition 2.3. (Rosenfeld [4]). The fuzzy subgroup M generated by the fuzzy set A is defined as the least fuzzy subgroup which _ A, denoted as M = [A].
a = b, then [A]/B = [A/B] (A/B is the fuzzy set of G/B by Definition 2.3)
Proof. By Proposition 2.4
Proposition 2.4 (Zhu [3]). [A] = OA,, • ~ F, A, is fuzzy subgroup of G. Let B be a fuzzy normal subgroup of G, in [5]. Wu has proven G/B = {xB: x s G } is a group. Let A be a fuzzy subgroup of G, we define a fuzzy set on G/B:
A/B: G/B ~ [0, 1], A/B(aB) = sup A(x). xB=aB
Proposition 2.5 (Chen [1]). The above A/B is a fuzzy subgroup of G/B.
Definition 2.6 (Chen [1]). We call the above A/B the fuzzy factor group of A with respect to B.
A, is the fuzzy subgroup of G, ~ F ; [A/B]=
0
Wa,
A/B c_ Wt;
Wt~ is the fuzzy subgroup of G/B, fl~ F'. It is clear that A/B c A~/B for any • e F, that is to say A, is one of the Wa, so
N (A,/B)=_ N
A A,
A/B W B
For any W,, let 7 : G ---, G/B, 7(x) = xB. It is clear that 7 is a homomorphism from G onto G/B, and 7(A,) = A,/B. For any x E G,
Proposition 2.7 (Chen and Gu [2]). Suppose A, B are fuzzy subgroups of G and G', respectively, with the sup property. Suppose A', B' are separately fuzzy normal subgroup of G and G', A'(e) = B'(e'). Then A x B/A' x B' ~- A/A' x B/B'.
= wp(7 (x)) = Wa(xB ) >~A/B(xB) >~A (x).
Proposition 2.8 (Zhu [3]). [A] = sup A(21 A .-. A A(2k), for any x eG, x l ... xk = x , k = l , 2 . . . . . -1 ~i is xi or xi
Proposition 2.9 (Chen [1]). Let A, A' be fuzzy subgroups of G, G', respectively, f ( A ) = A', where f is the homomorphism of G on G'. Let B be the fuzzy normal subgroup of G satisfying Gn = {x: x ~ G, B(x) = B(e)} ~ kerf, then A/B ~- A'.
Definition 2.10 (Chen and Gu [2]). Let A, B be
Hence A c_ 7 I(Wa), A/B c_ ?-I(W~)/B = W a, that is to say Wp is one of the A~JB, so ("] (AJB) = AA,
0
W p = [A/B].
A/BW~
For any aB ~ G/B, [A/B] (aB) = ( ~ (A,/B)) (aB) = inf supA~(x) xB=aB
fuzzy sets of the nonempty sets G, G'. The direct product A × B is the fuzzy subset of G x G' defined as following:
~> sup inf A,(x)
A x B(x, y) = min{ A(x), B(y)}, (x, y)e G x G'.
--- [AJ/B(aB).
3. Properties of the generating fuzzy factor group
xB = aB
= ((0 A,)/B)(aB)
Hence [A]/B ~ [A/B]. If B satisfies aB = bB follows a = b, it is clear that [A/B] = [A]/B. []
Proposition 3.1. Let A be the fuzzy subset of the group G, B a fuzzy normal subgroup of G, then [A]/B ~_ [A/B]. If B satisfies: aB = bB follows
Proposition 3.2. Let A, B be the fuzzy subsets of G, G', respectively, then [A] x [B] = [A x B].
D.-G. Chem. W . - X Gu / Fuzzy Sets and Systems 82 (1996) 3 5 7 - 3 6 0
Proof. [A x B] = N Cs, A x B ~_ C,, C, is the fuzzy subgroup of G x G', ct e r . [A] x [B] = NAp x ABe, A ~ Aa, B ~ B~, Ap, B~ are fuzzy subgroups of G, G', respectively, fle F', 7 ~ F". For any (x, y) e G x G',
Sincef(A) ~_ Kp, A ~ f - l ( K a ) , that is to say Kp is one of thef(A~), so n K a ___Nf(A,). Hence If(A)] (y) = ( n Ka) (y) = ( n f(A,)) (y)
(nAa x nBv)(x, y) = (AAa(x)) A (ABe(y)),
= inf sup As(x) f(x) = y
(NAa x B~)(x, y) =/~ (Aa(x) /~ B~(y)).
>/sup inf As(x)
So
f(x) = y
[A] x [B] = (NAa x OB~) = N(Aa x e~). It is clear A x B _c A~ x B~, that is to say Ao x B~ is one of the C, so [A] x [B] ~_ [A x B]. For any Cs, let C~(x)=C,(x,e'), C2(y')= C,(e, y'), then it is easy to prove C~ and C 2 are fuzzy subgroups of G,G', respectively, and C, ___C~ x C~, that is to say C, is one of the A~ x fl,, so [A] x [B] ~_ [ A x B]. Hence [A x B ] = [A] x [B]. []
Corollary 3.3. Let A', B' be fuzzy normal subgroups of G x G', then [A x B]/A' × B' = [A] x [B]/A' x B'.
Proposition 3.4. If A has the sup property, then [A] has it. By Proposition 2.8 it is clear.
Proposition 3.5. Let A, B be fuzzy subgroups of G, G', respectively, with the sup property, A', B' be fuzzy normal subgroups of G, G', respectively, A'(e) = B'(e'), then [A × B]/A' × B' ~ [A]/A' x [B]/B'. Proposition 3.6. Let f be the homomorphism from G onto G', A the fuzzy subset of G, then f([A]) _ [f(A)]. lf f is an isomorphism, then f ([A ] ) = [f(A)].
= f ( [ A ] ) (y). Hencef([A]) ~ If(A)]. If f is an isomorphism, then [f(A)] =f([A]).
4. The first fundamental theorem of isomorphism Theorem 4.1. Let t1 be the homomorphism from G onto G', A a fuzzy subgroup of G, A' = ~I(A). Let B be a ~l-invarant fuzzy normal subgroup of G, B' = ~I(B) then A/B ~ A'/B'. Proof. Let 7': G' ~ G'/B', 7'(x') = x'B'. It is clear that 7' is a homomorphism and 7'(A') = A'/B'. Hence A'/B' = (';',7) (A). By Proposition 2.9 we only need to prove GB = ker(7't/). B(e)=B'(e') is clear. For any xeG~, B(x) = B(e), B'(~/(x)) =
Hence rl(x)B' = B',(7'q)(x ) = B'. Hence x e ker(7'r/). Let xeker(7'r/), then (7'11) (x) = B', rl(x)B' = B', So
B'(rl(x)) = B'(e') = B(e).
sup
n f(x) = Y
S(y) = B(x) = B(e) = B'(e').
sup n (r) = ~ ( x )
Hence
Proof. For any y e G', f([A])(y)=f(
359
A~)(y)=supinfAs(x), f(x) = y
[f(A)] (y) = n K a ( y ) ,
A ~_ A,, f(A) ~ Kp, ~ e F, fle F', As and Ka be the fuzzy subgroup of G, G', respectively. Since A ~_ A,,f(A) ~_f(A,), that is to sayf(A,) is one of the Ks, so N f(A,) ~_ NKa.
B(y)=
y ~ ~/ ~(t/(x))
sup
B(y)=B(e).
~/(y) = ~/(x)
In case of B being q-invarant, we know
B(x) = B(y) by t/(x) = r/(y). Hence B(e)=
sup ~ ( x ) = n(y)
B(y)=B(x).
360
D.-G. Chen, W.-X. Gu / Fuzzy Sets and Systems 82 (1996) 357-360
H e n c e x e Gs a n d Gg = ker(v'r/). Hence A / B ~ A'/B'.
[]
References [1] D.-G. Chen, The fuzzy factor groups and the fundamental theorem of homomorphisms, J. Northeast Normal Univ. 2 (1994) 27-29.
[2] D.-G. Chen and W.-X. Gu, Product structure of the fuzzy factor groups, Fuzzy Sets and Systems 60 (1993) 229-232. [3] N.-D. Zhu, Generating fuzzy subgroups, Fuzzy Math. 2 (1989) 24-31. [4] A. Rosenfeld,Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. I-5] W.-M. Wu, Normal fuzzy subgroups, Fuzzy Math. 1 (1981) 21-30. I6] Y.-K. Zheng and W.-J. Liu, Some characterizations of the direct product of fuzzy subgroups, Fuzzy Systems Math. 2 (1989) 32-37. 1-7] L.A. Zadeh, Fuzzy sets, lnform, and Contro18(1965) 338-353.
sets and systems ELSEVIER
Fuzzy Sets and Systems 82 (1996) 357-360
Generating fuzzy factor groups and the fundamental theorem of isomorphism De-Gang Chen a'*, Wen-Xiang Gu b =Department of mathematics, JinZhou Teacher's College, .linZhou, LiaoNing, 121003, Peoples' Republic of China bDepartment of computer science, Northeast Normal University, ChangChun, JiLin, 130024, Peoples' Republic of China Received July 1994; revised June 1995
Abstract In this paper we study how a fuzzy set can generate a fuzzy factor group, we also study the product structure of the generating fuzzy factor group. At the end of this paper we prove the first fundamental theorem of isomorphisms of fuzzy groups.
Keywords: Fuzzy subgroup; Generating fuzzy group; Fuzzy factor group; Fundamental theorem of homomorphisms
1. Introduction
Since Rosenfeld gave the concept of fuzzy subgroup in his pioneering paper [4] in 1971, many mathematicians around the world have since studied deeply the properties of fuzzy groups. Research in fuzzy groups have produced a large number of papers. The references [1-6] represent some of the more recent works. In [1, 2] the concept of fuzzy factor group is given and the fundamental theorem of homomorphisms of fuzzy groups is proven, the product structure of fuzzy factor groups is also discussed. For the first time Zhu Nan-De introduced the concept of generating fuzzy subgroup and studied how a general fuzzy set could generate a fuzzy group. Since we have the definition of fuzzy
*Corresponding author.
factor group, we study the properties when a fuzzy set generate a fuzzy factor group. By use of the fundamental theorem of homomorphisms of fuzzy groups which has been proved in [1], we prove the first fundamental theorem of isomorphisms of fuzzy groups.
2. Preliminaries
Definition 2.1 (Rosenfeld [4]). Let A, B be fuzzy sets of G, if for any x, y E G, (1) A(xy)/> min{A(x), A(y)};
(2) a(x-1)/> a(x), then A is called a fuzzy subgroup of G. Definition 2.2 (Zadeh [7]). Let A, B be fuzzy sets of G, if for any x E G, A(x) .~ B(x), then we call A is contained by B, denoted as A _ B.
0165-0114/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 5 ) 0 0 2 7 1 - 5
358
D.-G. Chen, W.-X. Gu / Fuzzy Sets and Systems 82 (1996) 3 5 7 - 3 6 0
Definition 2.3. (Rosenfeld [4]). The fuzzy subgroup M generated by the fuzzy set A is defined as the least fuzzy subgroup which _ A, denoted as M = [A].
a = b, then [A]/B = [A/B] (A/B is the fuzzy set of G/B by Definition 2.3)
Proof. By Proposition 2.4
Proposition 2.4 (Zhu [3]). [A] = OA,, • ~ F, A, is fuzzy subgroup of G. Let B be a fuzzy normal subgroup of G, in [5]. Wu has proven G/B = {xB: x s G } is a group. Let A be a fuzzy subgroup of G, we define a fuzzy set on G/B:
A/B: G/B ~ [0, 1], A/B(aB) = sup A(x). xB=aB
Proposition 2.5 (Chen [1]). The above A/B is a fuzzy subgroup of G/B.
Definition 2.6 (Chen [1]). We call the above A/B the fuzzy factor group of A with respect to B.
A, is the fuzzy subgroup of G, ~ F ; [A/B]=
0
Wa,
A/B c_ Wt;
Wt~ is the fuzzy subgroup of G/B, fl~ F'. It is clear that A/B c A~/B for any • e F, that is to say A, is one of the Wa, so
N (A,/B)=_ N
A A,
A/B W B
For any W,, let 7 : G ---, G/B, 7(x) = xB. It is clear that 7 is a homomorphism from G onto G/B, and 7(A,) = A,/B. For any x E G,
Proposition 2.7 (Chen and Gu [2]). Suppose A, B are fuzzy subgroups of G and G', respectively, with the sup property. Suppose A', B' are separately fuzzy normal subgroup of G and G', A'(e) = B'(e'). Then A x B/A' x B' ~- A/A' x B/B'.
= wp(7 (x)) = Wa(xB ) >~A/B(xB) >~A (x).
Proposition 2.8 (Zhu [3]). [A] = sup A(21 A .-. A A(2k), for any x eG, x l ... xk = x , k = l , 2 . . . . . -1 ~i is xi or xi
Proposition 2.9 (Chen [1]). Let A, A' be fuzzy subgroups of G, G', respectively, f ( A ) = A', where f is the homomorphism of G on G'. Let B be the fuzzy normal subgroup of G satisfying Gn = {x: x ~ G, B(x) = B(e)} ~ kerf, then A/B ~- A'.
Definition 2.10 (Chen and Gu [2]). Let A, B be
Hence A c_ 7 I(Wa), A/B c_ ?-I(W~)/B = W a, that is to say Wp is one of the A~JB, so ("] (AJB) = AA,
0
W p = [A/B].
A/BW~
For any aB ~ G/B, [A/B] (aB) = ( ~ (A,/B)) (aB) = inf supA~(x) xB=aB
fuzzy sets of the nonempty sets G, G'. The direct product A × B is the fuzzy subset of G x G' defined as following:
~> sup inf A,(x)
A x B(x, y) = min{ A(x), B(y)}, (x, y)e G x G'.
--- [AJ/B(aB).
3. Properties of the generating fuzzy factor group
xB = aB
= ((0 A,)/B)(aB)
Hence [A]/B ~ [A/B]. If B satisfies aB = bB follows a = b, it is clear that [A/B] = [A]/B. []
Proposition 3.1. Let A be the fuzzy subset of the group G, B a fuzzy normal subgroup of G, then [A]/B ~_ [A/B]. If B satisfies: aB = bB follows
Proposition 3.2. Let A, B be the fuzzy subsets of G, G', respectively, then [A] x [B] = [A x B].
D.-G. Chem. W . - X Gu / Fuzzy Sets and Systems 82 (1996) 3 5 7 - 3 6 0
Proof. [A x B] = N Cs, A x B ~_ C,, C, is the fuzzy subgroup of G x G', ct e r . [A] x [B] = NAp x ABe, A ~ Aa, B ~ B~, Ap, B~ are fuzzy subgroups of G, G', respectively, fle F', 7 ~ F". For any (x, y) e G x G',
Sincef(A) ~_ Kp, A ~ f - l ( K a ) , that is to say Kp is one of thef(A~), so n K a ___Nf(A,). Hence If(A)] (y) = ( n Ka) (y) = ( n f(A,)) (y)
(nAa x nBv)(x, y) = (AAa(x)) A (ABe(y)),
= inf sup As(x) f(x) = y
(NAa x B~)(x, y) =/~ (Aa(x) /~ B~(y)).
>/sup inf As(x)
So
f(x) = y
[A] x [B] = (NAa x OB~) = N(Aa x e~). It is clear A x B _c A~ x B~, that is to say Ao x B~ is one of the C, so [A] x [B] ~_ [A x B]. For any Cs, let C~(x)=C,(x,e'), C2(y')= C,(e, y'), then it is easy to prove C~ and C 2 are fuzzy subgroups of G,G', respectively, and C, ___C~ x C~, that is to say C, is one of the A~ x fl,, so [A] x [B] ~_ [ A x B]. Hence [A x B ] = [A] x [B]. []
Corollary 3.3. Let A', B' be fuzzy normal subgroups of G x G', then [A x B]/A' × B' = [A] x [B]/A' x B'.
Proposition 3.4. If A has the sup property, then [A] has it. By Proposition 2.8 it is clear.
Proposition 3.5. Let A, B be fuzzy subgroups of G, G', respectively, with the sup property, A', B' be fuzzy normal subgroups of G, G', respectively, A'(e) = B'(e'), then [A × B]/A' × B' ~ [A]/A' x [B]/B'. Proposition 3.6. Let f be the homomorphism from G onto G', A the fuzzy subset of G, then f([A]) _ [f(A)]. lf f is an isomorphism, then f ([A ] ) = [f(A)].
= f ( [ A ] ) (y). Hencef([A]) ~ If(A)]. If f is an isomorphism, then [f(A)] =f([A]).
4. The first fundamental theorem of isomorphism Theorem 4.1. Let t1 be the homomorphism from G onto G', A a fuzzy subgroup of G, A' = ~I(A). Let B be a ~l-invarant fuzzy normal subgroup of G, B' = ~I(B) then A/B ~ A'/B'. Proof. Let 7': G' ~ G'/B', 7'(x') = x'B'. It is clear that 7' is a homomorphism and 7'(A') = A'/B'. Hence A'/B' = (';',7) (A). By Proposition 2.9 we only need to prove GB = ker(7't/). B(e)=B'(e') is clear. For any xeG~, B(x) = B(e), B'(~/(x)) =
Hence rl(x)B' = B',(7'q)(x ) = B'. Hence x e ker(7'r/). Let xeker(7'r/), then (7'11) (x) = B', rl(x)B' = B', So
B'(rl(x)) = B'(e') = B(e).
sup
n f(x) = Y
S(y) = B(x) = B(e) = B'(e').
sup n (r) = ~ ( x )
Hence
Proof. For any y e G', f([A])(y)=f(
359
A~)(y)=supinfAs(x), f(x) = y
[f(A)] (y) = n K a ( y ) ,
A ~_ A,, f(A) ~ Kp, ~ e F, fle F', As and Ka be the fuzzy subgroup of G, G', respectively. Since A ~_ A,,f(A) ~_f(A,), that is to sayf(A,) is one of the Ks, so N f(A,) ~_ NKa.
B(y)=
y ~ ~/ ~(t/(x))
sup
B(y)=B(e).
~/(y) = ~/(x)
In case of B being q-invarant, we know
B(x) = B(y) by t/(x) = r/(y). Hence B(e)=
sup ~ ( x ) = n(y)
B(y)=B(x).
360
D.-G. Chen, W.-X. Gu / Fuzzy Sets and Systems 82 (1996) 357-360
H e n c e x e Gs a n d Gg = ker(v'r/). Hence A / B ~ A'/B'.
[]
References [1] D.-G. Chen, The fuzzy factor groups and the fundamental theorem of homomorphisms, J. Northeast Normal Univ. 2 (1994) 27-29.
[2] D.-G. Chen and W.-X. Gu, Product structure of the fuzzy factor groups, Fuzzy Sets and Systems 60 (1993) 229-232. [3] N.-D. Zhu, Generating fuzzy subgroups, Fuzzy Math. 2 (1989) 24-31. [4] A. Rosenfeld,Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. I-5] W.-M. Wu, Normal fuzzy subgroups, Fuzzy Math. 1 (1981) 21-30. I6] Y.-K. Zheng and W.-J. Liu, Some characterizations of the direct product of fuzzy subgroups, Fuzzy Systems Math. 2 (1989) 32-37. 1-7] L.A. Zadeh, Fuzzy sets, lnform, and Contro18(1965) 338-353.