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On product of fuzzy subgroups
FUIIY
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 181-183
Short Communication
On product of fuzzy subgroups Asok Kumer Ray
Department of Mathematics, Dibrugarh University, Dibrugarh-786 004, Assam, India Received December 1997; received in revised form September 1998
Abstract In this short communication, we have studied some properties of the product of two fuzzy subsets and fuzzy subgroups. ~) 1999 Elsevier Science B.V. All rights reserved.
Keywords: Fuzzy subsets; Product of fuzzy subsets; Level subset; Fuzzy subgroup; Fuzzy normal subgroup; Fuzzy conjugate subgroup
1. Introduction Here we recall some basic and well-known definitions and properties.
Definition 1.1 (Novak [5]). Let A and B be fuzzy subsets of the sets G and H, respectively. The product of A and B, denoted by A × B, is the function defined by setting for all x in G and y in H,
(A × B)(x, y) = min(A(x),B(y)) Definition 1.2 (Rosenfeld [1]). A fuzzy subsetA of a group G is said to be a fuzzy subgroup of G if for all
Proof of the following theorem is a routine work.
Theorem 1.3. IrA and B are fuzzy subgroups of the groups G and H, respectively, then A x B is a fi~zzy subgroup of G x H. Definition 1.4 (Zadeh [2]). Let A be a fuzzy subset of a set S and let t E [0, 1]. The set At = {x E S/A(x) >~t} is called a level subset of A. Definition 1.5 (Mukherjee and Bhattacharya [3]). A fuzzy subgroup A of a group G is called fuzzy normal if for all x, y in G it fulfils the following condition:
x, y i n G (1) A(xy)>~ min(A(x),A(y)) and (2) A(x-1)>~A(x),
A(xy)=A(yx).
where the product ofx and y is denoted by xy and the inverse of x by x - 1.
fuzzy subgroup A of a group G is said to be conjugate to a fuzzy subgroup B of G if there exists x in G such that for all g in G
It is well known and easy to see that a fuzzy subgroup A of a group G satisfies A(x)<~A(e) and A(x -I ) = A ( x ) for all x E G, where e is the neutral element of G.
A(g) =B(x-lgx).
Definition 1.6 (Mukherjee and Bhattacharya [4]). A
2. Here we state some results whose proofs are very simple.
0165-0114/99/$ - see front matter (~ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00411-4
182
A.K. Ray~Fuzzy Sets and Systems 105 (1999) 181-183
Theorem 2.1. Let A and B be fuzzy subsets of the sets G and H, respectively, and let t E [0, 1]. Then (A × B)t =At × Bt.
such that A(a)>B(e') and B(b)>A(e). We have
(A × B)(a, b) = min(A(a),B(b)) > min(A(e),B(e'))
Theorem 2.2. Let A and B be fuzzy subgroups of the groups G and H, respectively, l f A and B are fuzzy normal then A × B is fuzzy normal. Theorem 2.3. Let a fuzzy subgroup A of a group G be conjugate to a fuzzy subgroup ~ of G and a fuzzy subgroup B of a group H be conjugate to a fuzzy subgroup fl of H. Then the fuzzy subgroup A x B of the group G × H is conjugate to the fuzzy subgroup ~×flofGxH. LetA and B be fuzzy subsets o f the groups G and H , respectively. If A × B is a fuzzy subgroup of G × H, then it is not necessary that both A and B should be fuzzy subgroups of G and H, respectively. Consider the following example: Let G = {e, a} where a 2 = e, and let H be the Klein four-group and, H = {et,x, y, xy} where x 2 = e ~ = y2 and xy = yx. Then G x H = {(e, e')}, (e,x), (e, y),
(e, xy), (a,e'), (a,x), (a, xy)}. Let A = {(e, 0.7), (a, 0.6)} and B = {(e', 0.9), (x, 1), (y, 0.8), (xy, 0.7)}. ThenA x B = {((e, e'), 0.7), ((e,x), 0.7),((e,y),O.7),((e, xy), 0.7), ((a, e'), 0.6), ((a, x), 0.6), ((a, y), 0.6), ((a, xy), 0.6)}. Here A x B is a fuzzy subgroup of G × H , whereas A is a fuzzy subgroup of G but B is not a fuzzy subgroup of H because B0.8 is not a subgroup o f H. 3. Here we shall state and prove some theorems.
= (A × B)(e,e'). Thus A × B is not a fuzzy subgroup of G × H. Hence either B(e')>~A(x) for all x in G or A(e)>~B(y) for all y in H. This completes the proof. []
Theorem 3.2. Let A and B be fuzzy subsets of the groups G and H, respectively, such that A(x) <~B(e') for all x in G, e' being the neutral element of H. I f A × B is a fuzzy subgroup ofG × 14, then A is a fuzzy subgroup of G. Proof. Let A × B be a fuzzy subgroup of G × H and x, y E G. Then (x, et),(y,e')E G x H. Now, using the property A(x) <~B(e~) for all x E G, we get,
A(xy) = min(A(xy), B(e~e') ) = (A × B)(x,e')(y,e') ~> min((A x B)(x,e~),(A x B)(y,e~)) = min(min(A(x), B(e')), min(A(y), B(e')))
= min(A(x),A(y)). Also,
A(x -l ) = min(A(x - l ),B(e ' - I )) = (A × B ) ( x - l , e ' - l ) = (A × B)(x,e') -l >>.(A x B)(x,e')
Theorem 3.1. Let A and B be fuzzy subsets of the
groups G and H, respectively. Suppose that e and e ~ are the neutral elements of G and H, respectively. I f A × B is a fuzzy subgroup ofG × 1-1,then at least one of the following two statements must hold. (1) B(e')>~A(x) for allx in G, (2) A(e)>~B(y) for all y in H. Proof. Let A × B be a fuzzy subgroup of G × H. By contraposition, suppose that none of the statements (1) and (2) holds. Then we can find a in G and b in H
= min(A(x), B(e')) = A(x). Hence A is a fuzzy subgroup of G. This completes the proof. [] By symmetry we can prove:
Theorem 3.3. Let A and B be fuzzy subsets of the groups G and H, respectively, such that B(x) <~A(e) for all x E H, e being the neutral element of G. I f A × B is a fuzzy subgroup ofG x H, then B is a fuzzy subgroup of H.
A.K. Ray~Fuzzy Sets and Systems 105 (1999) 181-183
From Theorems 3.1, 3.2 and 3.3 we have the following corollary. Corollary 3.4. Let A and B be fuzzy subsets of the oroups G and 11, respectively. I f A x B is a fuzzy subyroup of G x H, then either A is a fuzzy subgroup of G or B is a fuzzy subyroup of H.
Acknowledgements The author wishes to express his sincere gratitude to the referee for his valuable comments and advice.
183
References [1] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [2] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [3] N.P. Mukherjee, P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [4] N.P. Mukherjee, P. Bhattacharya, Fuzzy groups: some group theoretic analogs, lnform. Sci. 39 (1986) 247-268. [5] V. Novak, Fuzzy Sets and Their Applications, Adam Hilger, Bristol, 1989.
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 181-183
Short Communication
On product of fuzzy subgroups Asok Kumer Ray
Department of Mathematics, Dibrugarh University, Dibrugarh-786 004, Assam, India Received December 1997; received in revised form September 1998
Abstract In this short communication, we have studied some properties of the product of two fuzzy subsets and fuzzy subgroups. ~) 1999 Elsevier Science B.V. All rights reserved.
Keywords: Fuzzy subsets; Product of fuzzy subsets; Level subset; Fuzzy subgroup; Fuzzy normal subgroup; Fuzzy conjugate subgroup
1. Introduction Here we recall some basic and well-known definitions and properties.
Definition 1.1 (Novak [5]). Let A and B be fuzzy subsets of the sets G and H, respectively. The product of A and B, denoted by A × B, is the function defined by setting for all x in G and y in H,
(A × B)(x, y) = min(A(x),B(y)) Definition 1.2 (Rosenfeld [1]). A fuzzy subsetA of a group G is said to be a fuzzy subgroup of G if for all
Proof of the following theorem is a routine work.
Theorem 1.3. IrA and B are fuzzy subgroups of the groups G and H, respectively, then A x B is a fi~zzy subgroup of G x H. Definition 1.4 (Zadeh [2]). Let A be a fuzzy subset of a set S and let t E [0, 1]. The set At = {x E S/A(x) >~t} is called a level subset of A. Definition 1.5 (Mukherjee and Bhattacharya [3]). A fuzzy subgroup A of a group G is called fuzzy normal if for all x, y in G it fulfils the following condition:
x, y i n G (1) A(xy)>~ min(A(x),A(y)) and (2) A(x-1)>~A(x),
A(xy)=A(yx).
where the product ofx and y is denoted by xy and the inverse of x by x - 1.
fuzzy subgroup A of a group G is said to be conjugate to a fuzzy subgroup B of G if there exists x in G such that for all g in G
It is well known and easy to see that a fuzzy subgroup A of a group G satisfies A(x)<~A(e) and A(x -I ) = A ( x ) for all x E G, where e is the neutral element of G.
A(g) =B(x-lgx).
Definition 1.6 (Mukherjee and Bhattacharya [4]). A
2. Here we state some results whose proofs are very simple.
0165-0114/99/$ - see front matter (~ 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00411-4
182
A.K. Ray~Fuzzy Sets and Systems 105 (1999) 181-183
Theorem 2.1. Let A and B be fuzzy subsets of the sets G and H, respectively, and let t E [0, 1]. Then (A × B)t =At × Bt.
such that A(a)>B(e') and B(b)>A(e). We have
(A × B)(a, b) = min(A(a),B(b)) > min(A(e),B(e'))
Theorem 2.2. Let A and B be fuzzy subgroups of the groups G and H, respectively, l f A and B are fuzzy normal then A × B is fuzzy normal. Theorem 2.3. Let a fuzzy subgroup A of a group G be conjugate to a fuzzy subgroup ~ of G and a fuzzy subgroup B of a group H be conjugate to a fuzzy subgroup fl of H. Then the fuzzy subgroup A x B of the group G × H is conjugate to the fuzzy subgroup ~×flofGxH. LetA and B be fuzzy subsets o f the groups G and H , respectively. If A × B is a fuzzy subgroup of G × H, then it is not necessary that both A and B should be fuzzy subgroups of G and H, respectively. Consider the following example: Let G = {e, a} where a 2 = e, and let H be the Klein four-group and, H = {et,x, y, xy} where x 2 = e ~ = y2 and xy = yx. Then G x H = {(e, e')}, (e,x), (e, y),
(e, xy), (a,e'), (a,x), (a, xy)}. Let A = {(e, 0.7), (a, 0.6)} and B = {(e', 0.9), (x, 1), (y, 0.8), (xy, 0.7)}. ThenA x B = {((e, e'), 0.7), ((e,x), 0.7),((e,y),O.7),((e, xy), 0.7), ((a, e'), 0.6), ((a, x), 0.6), ((a, y), 0.6), ((a, xy), 0.6)}. Here A x B is a fuzzy subgroup of G × H , whereas A is a fuzzy subgroup of G but B is not a fuzzy subgroup of H because B0.8 is not a subgroup o f H. 3. Here we shall state and prove some theorems.
= (A × B)(e,e'). Thus A × B is not a fuzzy subgroup of G × H. Hence either B(e')>~A(x) for all x in G or A(e)>~B(y) for all y in H. This completes the proof. []
Theorem 3.2. Let A and B be fuzzy subsets of the groups G and H, respectively, such that A(x) <~B(e') for all x in G, e' being the neutral element of H. I f A × B is a fuzzy subgroup ofG × 14, then A is a fuzzy subgroup of G. Proof. Let A × B be a fuzzy subgroup of G × H and x, y E G. Then (x, et),(y,e')E G x H. Now, using the property A(x) <~B(e~) for all x E G, we get,
A(xy) = min(A(xy), B(e~e') ) = (A × B)(x,e')(y,e') ~> min((A x B)(x,e~),(A x B)(y,e~)) = min(min(A(x), B(e')), min(A(y), B(e')))
= min(A(x),A(y)). Also,
A(x -l ) = min(A(x - l ),B(e ' - I )) = (A × B ) ( x - l , e ' - l ) = (A × B)(x,e') -l >>.(A x B)(x,e')
Theorem 3.1. Let A and B be fuzzy subsets of the
groups G and H, respectively. Suppose that e and e ~ are the neutral elements of G and H, respectively. I f A × B is a fuzzy subgroup ofG × 1-1,then at least one of the following two statements must hold. (1) B(e')>~A(x) for allx in G, (2) A(e)>~B(y) for all y in H. Proof. Let A × B be a fuzzy subgroup of G × H. By contraposition, suppose that none of the statements (1) and (2) holds. Then we can find a in G and b in H
= min(A(x), B(e')) = A(x). Hence A is a fuzzy subgroup of G. This completes the proof. [] By symmetry we can prove:
Theorem 3.3. Let A and B be fuzzy subsets of the groups G and H, respectively, such that B(x) <~A(e) for all x E H, e being the neutral element of G. I f A × B is a fuzzy subgroup ofG x H, then B is a fuzzy subgroup of H.
A.K. Ray~Fuzzy Sets and Systems 105 (1999) 181-183
From Theorems 3.1, 3.2 and 3.3 we have the following corollary. Corollary 3.4. Let A and B be fuzzy subsets of the oroups G and 11, respectively. I f A x B is a fuzzy subyroup of G x H, then either A is a fuzzy subgroup of G or B is a fuzzy subyroup of H.
Acknowledgements The author wishes to express his sincere gratitude to the referee for his valuable comments and advice.
183
References [1] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [2] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [3] N.P. Mukherjee, P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [4] N.P. Mukherjee, P. Bhattacharya, Fuzzy groups: some group theoretic analogs, lnform. Sci. 39 (1986) 247-268. [5] V. Novak, Fuzzy Sets and Their Applications, Adam Hilger, Bristol, 1989.