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The homomorphic image of a fuzzy subgroup is always a fuzzy subgroup
FuzzySets and Systems33 (1989)255-256 North-Holland
255
SHORT COMMUNICATION H O M O M O I ~ P E ~ C I M A G E OF A F U Z Z Y S U B G R O U P IS A L W A Y S A F U Z Z Y S U B G R O U P Mehmet $ait E R O ~ L U Departmentof Matb.emaffcs, Earadeniz TechnicalUniversity, Trabzon, Turkey ReceivedNovember1988 Abstract: Let L be a completelydistributive,completelatticewithuniversalbound0 and 1, and X any group. We willprove that the imageof a fuzzygroup/A:X--*/.,under a homomorphism f :X--~Y is alwaysa fuzzygroup withoutany restrictionon/~ and f. Keywovds:Fuzzygroup, homomorphicimage. L Inb'oduction In [3], Rosenfeld proved that the homomorphic image of a fuzzy subgroup : X--~ I with sup property is a fuzzy subgroup. Here, I is the closed unit hlterval [0,1] in R. Anthony and Sherwood [1] PrOVed that if ~ : X - ~ I is a fuzzy subgroupoid with respect to a continuous t-norm T, specially if ~ is a fuzzy subgroupoid in the sense of Rosenfeld, then f ( ~ ) i.s a f~zzy subgroupoid on f ( X ) for every homomorphism f : X--~ Y. In [2], Liu tried to prove that if f : X--~ Y is a surjective homomorphism, then the homomorphic image f ( ~ ) of every fuzzy subgroup ~ :X--~ L is a fuzzy subgroup. Unfortunately his proof is not correct, his conclusion on page 136 line 2 and 3 is false. Now, without any restriction on the homomorphisms or fuzzy subgroups, we will prove that homomorphic images of fuzzy subgroups ~: X--~ L are fuzzy subgroups. 2. Homomo~pMe binge Let L be any complete and completely distributive lattice with universal bound 0 and 1. Defmll~n 1. Let X be a groupoid (or group). I~:X--~L is called an L-fuzzy groupoid (or L-fuzzy group) on X iff O) ~,(xy) ~ ~,(x) ^ ~,(y) Vx, y ~ x
(or (ii) ~(x -~) ~ ~(x) Vx ¢ X and (i)). T u o r e m 1. Let f : X - - , Y be a homomorphism between groupoid X and Y. For every fuzzy subgroupoid ~ in X, f(~) is a fuzzy subgroupoid in Y. 0165-0114/89/$3.50 ~) 1989,ElsevierSciencePublishersB.V. (North-Holland)
M.$. Ero~lu PrOOf. By definition,
f(~)(y): =
sup ~(x),
Vy ~ Y (sup ~: = 0).
We have to prove that f(~)(y~) >~f(~)(y) ^f(~)0~)
Vy, y ¢ Y.
(I)
For every y ¢ Y, we define Xy:-- f-1(y). Since f is a homomorphism,
x,x~x,~, V y , ~ Y
(2)
Let y, y ¢ Y be arbitrarily given. If yy f Ira(f) = f ( X ) , then by definition f ( ~ ) ( y y ) - 0. But if yy ~ f ( X ) , i.e., Xy~- f~, then by (2), Xy - ~ or X~ = ~; thus f(/~)(y) - 0 or f(/~)(~) -- 0. So we have, if yy ~ f ( X ) , f(~)(y~) = 0 = f ( ~ ) ( y ) ^ f(~)(~). Let X y ~ e . If X y = ~ or X ~ - e , then f ( i z ) ( y ) = 0 or f ( / ~ ) ( y ) = 0 and (1) is satisfied. If Xy ~ f, X~ ~ f, we get by (2), f ( ~ ) ( y y ) - - sup ~t(z)~ > sup /~(z)ffi sup/A(x.~) zeXr/ zcX~,Xy xeX, Hence by (i), ~x, ~exy
Thus
f(~)(yY) ~>f(~)(y) ^ f(~)(Y). One can define L-fuzzy ring, L-fuzzy algebra, L-fuzzy field on X in the same way.
Theorem 2. If X and Y are groups (rings, algebras, fields) and f :X--~ Y is a group- (ring-, algebra-, field-) homomorphism, then for every L-fuzzy group (L-fuzzy ring, L-fuzzy algebro., L-fuzzy field), on X, the image f(l~) is a L-fuzzy ~ro~p (L-fuzzy ring, L-fuzzy algebra, L-fuzzy field) on X. P r e ~ . By Theorem 1, for a fuzzy subgroup we need only to show that f(~)(y-1)>~f(~)(y) Vy ¢ Y. If y-1 ¢ f ( X ) then y ~ f ( X ) , and by definition f(/~)(y-~) = 0 = f ( ~ ) ( y ) . If y-X e l ( X ) , then y e f ( X ) and we have f(~)(y-~) f(/~)(y) as shown in [2], p. 137. Nothing more is to prove. Referen~
[I] J.M. Anthony and H. Shelwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [2] W.J. Liu, Fuzzyinvafiantsubgroups and fuzzyideals, Fuzzy Sets and Systems8 (1982) 133-139. [3] A. Re,enfold, Fuzzy groups,J. Math. Anal. Appl. 35 (1971) 512-517.
255
SHORT COMMUNICATION H O M O M O I ~ P E ~ C I M A G E OF A F U Z Z Y S U B G R O U P IS A L W A Y S A F U Z Z Y S U B G R O U P Mehmet $ait E R O ~ L U Departmentof Matb.emaffcs, Earadeniz TechnicalUniversity, Trabzon, Turkey ReceivedNovember1988 Abstract: Let L be a completelydistributive,completelatticewithuniversalbound0 and 1, and X any group. We willprove that the imageof a fuzzygroup/A:X--*/.,under a homomorphism f :X--~Y is alwaysa fuzzygroup withoutany restrictionon/~ and f. Keywovds:Fuzzygroup, homomorphicimage. L Inb'oduction In [3], Rosenfeld proved that the homomorphic image of a fuzzy subgroup : X--~ I with sup property is a fuzzy subgroup. Here, I is the closed unit hlterval [0,1] in R. Anthony and Sherwood [1] PrOVed that if ~ : X - ~ I is a fuzzy subgroupoid with respect to a continuous t-norm T, specially if ~ is a fuzzy subgroupoid in the sense of Rosenfeld, then f ( ~ ) i.s a f~zzy subgroupoid on f ( X ) for every homomorphism f : X--~ Y. In [2], Liu tried to prove that if f : X--~ Y is a surjective homomorphism, then the homomorphic image f ( ~ ) of every fuzzy subgroup ~ :X--~ L is a fuzzy subgroup. Unfortunately his proof is not correct, his conclusion on page 136 line 2 and 3 is false. Now, without any restriction on the homomorphisms or fuzzy subgroups, we will prove that homomorphic images of fuzzy subgroups ~: X--~ L are fuzzy subgroups. 2. Homomo~pMe binge Let L be any complete and completely distributive lattice with universal bound 0 and 1. Defmll~n 1. Let X be a groupoid (or group). I~:X--~L is called an L-fuzzy groupoid (or L-fuzzy group) on X iff O) ~,(xy) ~ ~,(x) ^ ~,(y) Vx, y ~ x
(or (ii) ~(x -~) ~ ~(x) Vx ¢ X and (i)). T u o r e m 1. Let f : X - - , Y be a homomorphism between groupoid X and Y. For every fuzzy subgroupoid ~ in X, f(~) is a fuzzy subgroupoid in Y. 0165-0114/89/$3.50 ~) 1989,ElsevierSciencePublishersB.V. (North-Holland)
M.$. Ero~lu PrOOf. By definition,
f(~)(y): =
sup ~(x),
Vy ~ Y (sup ~: = 0).
We have to prove that f(~)(y~) >~f(~)(y) ^f(~)0~)
Vy, y ¢ Y.
(I)
For every y ¢ Y, we define Xy:-- f-1(y). Since f is a homomorphism,
x,x~x,~, V y , ~ Y
(2)
Let y, y ¢ Y be arbitrarily given. If yy f Ira(f) = f ( X ) , then by definition f ( ~ ) ( y y ) - 0. But if yy ~ f ( X ) , i.e., Xy~- f~, then by (2), Xy - ~ or X~ = ~; thus f(/~)(y) - 0 or f(/~)(~) -- 0. So we have, if yy ~ f ( X ) , f(~)(y~) = 0 = f ( ~ ) ( y ) ^ f(~)(~). Let X y ~ e . If X y = ~ or X ~ - e , then f ( i z ) ( y ) = 0 or f ( / ~ ) ( y ) = 0 and (1) is satisfied. If Xy ~ f, X~ ~ f, we get by (2), f ( ~ ) ( y y ) - - sup ~t(z)~ > sup /~(z)ffi sup/A(x.~) zeXr/ zcX~,Xy xeX, Hence by (i), ~x, ~exy
Thus
f(~)(yY) ~>f(~)(y) ^ f(~)(Y). One can define L-fuzzy ring, L-fuzzy algebra, L-fuzzy field on X in the same way.
Theorem 2. If X and Y are groups (rings, algebras, fields) and f :X--~ Y is a group- (ring-, algebra-, field-) homomorphism, then for every L-fuzzy group (L-fuzzy ring, L-fuzzy algebro., L-fuzzy field), on X, the image f(l~) is a L-fuzzy ~ro~p (L-fuzzy ring, L-fuzzy algebra, L-fuzzy field) on X. P r e ~ . By Theorem 1, for a fuzzy subgroup we need only to show that f(~)(y-1)>~f(~)(y) Vy ¢ Y. If y-1 ¢ f ( X ) then y ~ f ( X ) , and by definition f(/~)(y-~) = 0 = f ( ~ ) ( y ) . If y-X e l ( X ) , then y e f ( X ) and we have f(~)(y-~) f(/~)(y) as shown in [2], p. 137. Nothing more is to prove. Referen~
[I] J.M. Anthony and H. Shelwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [2] W.J. Liu, Fuzzyinvafiantsubgroups and fuzzyideals, Fuzzy Sets and Systems8 (1982) 133-139. [3] A. Re,enfold, Fuzzy groups,J. Math. Anal. Appl. 35 (1971) 512-517.