Homomorphism and Anti-Homomorphism of an Intuitionistic Anti L-Fuzzy Translation

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1 International Journal of Computer & Organization rends Volume 5 Issue 2 March to pril 2015 Homomorphism and nti-homomorphism of an Intuitionistic nti L-Fuzzy ranslation 1 Dr. P.Pandiammal, 2 L.Vinotha 1 Department of Mathematics, PSNCE, Dindigul, amil Nadu, India 2 Department of Mathematics, PSNCE, Dindigul, amil Nadu, India BSRC his paper contains some definitions and results in intuitionistic anti L-fuzzy translation of intuitionistic anti L-fuzzy M-subgroup of a M-group, which are required in the sequel. Some properties of homomorphism and anti-homomorphism of intuitionistic anti L-fuzzy translation are also established. Keywords: L-fuzzy set, L-fuzzy M-subgroup, Homomorphism, nti-homomorphism, subgroups, intuitionistic L-fuzzy anti translation and subgroups. INRODUCION: he notion of fuzzy sets was introduced by L.. Zadeh [12]. Fuzzy set theory has been developed in many directions by many researchers and has evoked great interest among mathematicians working in different fields of mathematics, such as topological spaces, functional analysis, loop, group, ring, near ring, vector spaces, automation. In 1971, Rosenfield [1] introduced the concept of fuzzy subgroup. Motivated by this, many mathematicians started to review various concepts and theorems of abstract algebra in the broader frame work of fuzzy settings. In [2], Biswas introduced the concept of anti-fuzzy subgroups of groups. Palaniappan. N and Muthuraj, [8] defined the homomorphism, anti-homomorphism of a fuzzy and an anti-fuzzy subgroups. Pandiammal. P, Natarajan. R, and Palaniappan. N, [10] defined the homomorphism, anti-homomorphism of an anti L-fuzzy M-subgroup. Kandasamy[4] introduced the concept of fuzzy translation and fuzzy multiplication. he idea of fuzzy magnified translation has been introduced by Majumder and Sardar [5]. Pandiammal. P [11] defined the concept of Intuitionistic nti L- fuzzy M-subgroups. In this paper we define a new algebraic structure of intuitionistic anti L-fuzzy translation of intuitionistic anti L-fuzzy M-subgroup of an M-group and study some their related properties. 1. PRELIMINRIES: INUIIONISIC NI L-FUZZY M-SUBGROUPS 1.1 Definition: Let (G, ) be a M- group. n intuitionistic L-fuzzy subset of G is said to be an intuitionistic L-fuzzy M-subgroup (ILFMSG) of G if the following conditions are satisfied: (i) ( mxy ) (x) (y), (ii) ( x -1 ) ( x ), (iii) ( mxy ) (x) (y), (iv) ( x -1 ) ( x ), for all x and y in G. 1.2 Definition: n intuitionistic fuzzy subset μ in a group G is said to be an intuitionistic anti fuzzy subgroup of G if the following axioms are satisfied. (i) ( xy ) (x) (y), (ii) ( x -1 ) ( x ), (iii) ( xy ) (x) (y), (iv) ( x -1 ) ( x ), for all x & y in G. 1.3 Proposition: Let G be a group. n intuitionistic fuzzy subset μ in a group G is said to be an intuitionistic anti fuzzy subgroup of G if the following conditions are satisfied, (i) ( xy ) (x) (y), (ii) ( xy ) (x) (y), for all x,y in G. 1.4 Definition: Let G be an M-group and μ be an intuitionistic anti fuzzy group of G. If (mx) (x) and (mx) (x) for all x in G and m in M then μ is said to be an intuitionistic anti fuzzy subgroup with operator of G. We use the phrase μ intuitionistic anti L-fuzzy M-subgroup of G. ISSN: Page 14

2 International Journal of Computer & Organization rends Volume 5 Issue 2 March to pril Example: Let H be M-subgroup of an M- group G and let = (, ) be an intuitionistic fuzzy set in G defined by 0.3 ; x ε H (x) = 0.5; otherwise 0.6; x ε H (x) = 0.3; otherwise For all x in G. hen it is easy to verify that = (, ) anti-fuzzy M- subgroup of G. 1.6 Definition: Let be an intuitionistic L-fuzzy subset of X and and in [ 0, 1 Sup{ (x) + (x) : xx, 0 (x) (, ) + (x) 1 }]. hen = is called an intuitionistic L-fuzzy translation of if (x) = (x) = (x) +, (x) = (x) = (x) +, + 1 Sup{ (x) + (x) : xx, 0 (x) + (x) 1}, for all x in X. 2 PROPERIES OF INUIIONISIC NI L-FUZZY RNSLION: 2.1 heorem: If intuitionistic anti L- fuzzy translation of an intuitionistic anti L-fuzzy M-subgroup of a M-group G, then (x -1 ) = (x) and (x -1 ) = (x), (x) (e) and (x) (e), for all x and e in G. Proof: Let x and e be elements of G. Now, (x) = (x) + = ( (x -1 ) -1 ) + (x -1 ) + = ( x -1 ) = (x -1 ) + (x) + = (x). herefore, (x) = (x -1 ), for x in G. nd, (x) = (x)+ = ( (x -1 ) -1 )+ (x -1 )+ = (x -1 ) = (x -1 ) + (x) + = (x). herefore, (x) = (x -1 ), for x in G. Now, (e) = (e) + = (xx -1 ) + { (x) (x -1 ) }+ = (x) + = (x). herefore, (e) (x),for x in G. nd (e) = (e) + = (xx -1 ) + { (x) (x -1 ) }+ = (x) + = (x). herefore, (e) (x), for x in G. 2.2 heorem: If intuitionistic L- fuzzy translation of an intuitionistic L-fuzzy M-subgroup of a M-group G, then (i) (xy -1 ) = (e) implies (x) = (y), (ii) ( xy -1 ) = (e) implies (x) = (y), for all x, y and e in G. Proof: Let x, y and e be elements of G. Now, (x) = (x) + = (xy -1 y) + ( (xy -1 ) + ) ( (y) + ) = (xy -1 ) (y) = (e) (y) = (y) = (y)+ = (yx -1 x) + { (yx -1 ) (x) }+ = ( (yx -1 ) + ) ( (x) + ) = (yx -1 ) (x) = (e) (x) = (x). herefore, (x) = (y), for all x & y in G. nd (x) = (x) + = (xy -1 y) + { (xy -1 ) (y)}+ = ( (xy -1 )+ ) ( (y) + ) = (xy -1 ) (y) = (e) (y) = (y) = (y) + = (yx -1 x) + { (yx -1 ) ( (x) }+ = ( (yx -1 )+) ( (x) + ) = (yx -1 ) (x) = (e) (x) = (x). herefore, (x) = (y), for all x & y in G. 2.3 heorem: If intuitionistic anti L-fuzzy translation of an intuitionistic anti L-fuzzy M-subgroup of a M-group G, then subgroup of a M-group G, for all x and y in G. Proof: ssume that intuitionistic anti L-fuzzy translation of an intuitionistic anti L- fuzzy M-subgroup of a M-group G. Let x and y in G. We have ISSN: Page 15

3 International Journal of Computer & Organization rends Volume 5 Issue 2 March to pril 2015 (mxy -1 ) = (mxy -1 ) + { (x) (y -1 )}+ = { (x) (y) }+ = ( (x) + ) ( (y)+ ) = (x) (y). herefore, (mxy -1 ) (x) (y), for all x and y in G. nd (mxy -1 )= (mxy -1 ) + { (x) (y -1 ) }+ = { (x) (y) }+ = ( (x) + ) ( (y) + ) = (x) (y). herefore, (mxy -1 ) (x) (y), for all x and y in G. Hence intuitionistic anti L-fuzzy M-subgroup of a M-group G. 2.4 heorem: If intuitionistic anti L-fuzzy translation of an intuitionistic anti L- fuzzy M-subgroup of a M-group G, then H = { xg : (x) = (e) and (x) = (e) } is a M-subgroup of G. Proof: Let x, y and e be elements of G. Given H = { xg : (x) = (e) and (x) = (e) }. Now, (x -1 ) = ( x ) = (e) and (x -1 ) = (x) = (e). herefore, ( x -1 ) = (e) and ( x - 1 ) = (e). herefore, x -1 H. Now, ( xy -1 ) (x) (y) = (e) (e) = (e), and (e) = ( (xy -1 )(xy -1 ) -1 ) (xy -1 ) (xy -1 ) = (xy -1 ). herefore, (e) = (xy -1 ), for all x and y in G. Now, (xy -1 ) (x) (y) = (e) (e) = (e), and (e) = ( (xy -1 )(xy -1 ) -1 ) (xy -1 ) (xy -1 ) = (xy -1 ). herefore, (e) = (xy -1 ),for all x and y in G. herefore, xy -1 in H. Hence H M-subgroup of G. 2.5 heorem: Let be an intuitionistic anti L- fuzzy translation of an intuitionistic anti L-fuzzy M-subgroup of a M-group G. If (xy -1 ) = 0, then (x) = (y) and if (xy -1 ) = 1, then (x) = (y). Proof: Let x and y be elements of G. Now, (x) = ( xy -1 y) (xy -1 ) (y) = 0 (y)= (y) = (y -1 ) = ( x -1 xy -1 ) (x -1 ) (xy -1 ) = (x) (xy -1 ) = (x) 0 = (x). herefore, (x) = (y), for all x and y in G. Now, (x) = ( xy -1 y) (xy -1 ) (y) = 1 (y) = (y) = (y -1 ) = ( x -1 xy -1 ) (x -1 ) (xy -1 ) = (x) (xy -1 ) = (x) 1 = (x). herefore, (x) = (y), for all x and y in G. 2.6 heorem: Let G be a M-group. If subgroup of G, then (xy) = (x) (y) and (xy) = (x) (y), for each x and y in G with (x) (y) and (x) (y). Proof: Let x and y be elements of G. ssume that (x) (y) and (x) (y). hen, (y) = ( x -1 xy ) (x -1 ) (xy) = (x) (xy) = (xy) (x) (y) = (y). herefore, (xy) = (y)= (x) (y), for all x and y in G. hen, (y) = ( x -1 xy) (x -1 ) (xy) = (x) (xy) = (xy) (x) (y) = (y). herefore, (xy) = (y) = (x) (y), for all x and y in G. 2.7 heorem: Let (G, ) and (G, ) be any two M-groups. If f : G G is a homomorphism, then the homomorphic image of an intuitionistic anti L-fuzzy translation of an intuitionistic anti L-fuzzy M-subgroup of a M-group G ISSN: Page 16

4 International Journal of Computer & Organization rends Volume 5 Issue 2 March to pril 2015 subgroup of a M-group G. Proof: Let (G, ) and (G, ) be any two M-groups and f : G G be a homomorphism. hat is f(x y) = f(x) f(y), f(mx y) = mf(x) f(y), for all x and y in G and m in M. (, ) (, ) Let V = f( ), where subgroup of a M-group G. We have to prove that V intuitionistic anti L-fuzzy M-subgroup of a M-group G. Now, for f(x) and f(y) in G, we have V [ mf(x) ( f(y) -1 ) ] = V [mf(x) f(y -1 ) ] = V [f(mx y -1 )] ( mx y -1 ) = ( mx y -1 ) + { (x) ( y -1 ) } + { (x) (y)}+ = ( (x)+) ( (y)+) = ( x ) ( y ) which implies that V [ mf(x) ( f(y) -1 ) ] V ( f(x) ) V ( f(y) ), for all f(x) and f(y) in G. nd, V [ mf(x) (f(y) -1 )] = V [ mf(x) f(y -1 ) ] = V [ f(mx y -1 ) ] (mxy -1 ) = ( mx y -1 ) + { (x) ( y -1 ) } + { (x) ( y ) } + = ( (x)+) ( (y) + ) = (x) (y) which implies that V [ mf(x)(f(y) -1 )] V (f(x)) V ( f(y) ), for all f(x) and f(y) in G. herefore, V intuitionistic anti L-fuzzy M-subgroup of a M-group G. Hence the homomorphic image of an intuitionistic anti L-fuzzy translation of of G is an subgroup of a M-group G. 2.8 heorem: Let (G, ) and (G, ) be any two M-groups. If f : G G is a homomorphism, then the homomorphic pre-image of an intuitionistic anti L-fuzzy translation of an subgroup V of G is an subgroup of a M-group G. Proof: Let (G, ) and (G, ) be any two M-groups and f : G G be a homomorphism. hat is f(xy) = f(x) f(y), f(mx y) = mf(x) f(y), for all x and y in G and m in M. V (, ) V (, ) Let = = f(), where intuitionistic anti L-fuzzy translation of subgroup V of a M- group G. We have to prove that intuitionistic anti L-fuzzy M-subgroup of a M-group G. Let x and y in G. hen, (mx y -1 ) = ( f(mxy -1 ) ) = ( mf(x)f(y -1 ) = [ mf(x)( f(y) ) -1 ] = V [mf(x)(f(y)) 1 ]+ { V (f(x)) V (f(y))}+ =( V (f(x))+) ( V (f(y))+) = ( f(x) ) ( f(y) ) = (x) (y) (mx y -1 ) (x) (y), for all x and y in G. nd, (mx y -1 ) = ( f(mxy -1 ) ) = ( mf(x)f(y -1 ) ) = [mf(x)(f(y)) -1 ] = V [mf(x)(f(y)) -1 ]+ { V (f(x) ) V ( f(y))}+ = ( V (f(x))+)( V (f(y))+ ) = ( f(x) ) ( f(y) ) = (x) (y) (mx y -1 ) (x) (y), for all x and y in G. herefore, intuitionistic anti L-fuzzy M-subgroup of a M-group G. Hence the homomorphic pre-image of an subgroup V of a M-group G intuitionistic anti L-fuzzy M-subgroup of a M-group G. 2.9 heorem: Let (G, ) and (G, ) be any two M-groups. If f : G G antihomomorphism, then the antihomomorphic image(pre-image) of an M-subgroup of a M-group G M-subgroup of a M-group G. Proof: Let (G, ) and (G, ) be any two M-groups and f : G G be an antihomomorphism. hat is f(xy) = f(x)f(y), f(mxy) = mf(x) f(y), for all x and y in G and m in M. ISSN: Page 17

5 International Journal of Computer & Organization rends Volume 5 Issue 2 March to pril 2015 (, ) (, ) Let V = f( ), where M-subgroup of a M-group G. We have to prove that V intuitionistic anti L-fuzzy normal M-subgroup of a M-group G. Now, for f(x) and f(y) in G, clearly V intuitionistic anti L-fuzzy M-subgroup of a M-group G. We have, V ( mf(x) f(y) ) = V ( f(myx) ) (myx) = (myx) + = (mxy) + = (mxy) V ( f(mxy) ) = V ( mf(y) f(x) ) V ( mf(x)f(y) ) = V ( mf(y) f(x) ), for f(x) and f(y) in G. nd, V ( mf(x) f(y) ) = V ( f(myx) ) (myx) = (myx) + = (mxy) + = (mxy) V ( f(mxy) ) = V ( mf(y) f(x) ) V ( mf(x)f(y) ) = V ( mf(y) f(x) ), for f(x) and f(y) in G. herefore, V intuitionistic anti L-fuzzy normal M-subgroup of a M-group G. Hence the anti-homomorphic image of an intuitionistic anti L-fuzzy translation of of a M-group G intuitionistic anti L- fuzzy normal M-subgroup of a M- group G. REFERENCES: [1] zriel Rosenfeld, Fuzzy Groups, Journal of mathematical analysd applications 35, (1971) [2] R. Biswas, Fuzzy subgroups and anti-fuzzy subgroups, Fuzzy Sets and Systems 35 (1990), [3] J.. Goguen, L-fuzzy sets, J. Math. nal. ppl.18 (1967), [4] W.B.V. Kandasamy, Smrandache fuzzy algebra merican Research Press, (2003), pp [5] S.K. Majumder, S.K. Sardar, Fuzzy magnified translation on groups Journal of Mathematics, North Bengal University, 1(2), (2008), [6] Mohamed saad, Groups and Fuzzy Subgroups, Fuzzy Sets and Systems 39(1991) [7] Prabir Bhattacharya, Fuzzy subgroups: Some characterizations, J. Math. nal. ppl. 128 (1981) [8] N.Palaniappan, R.Muthuraj, he homomorphism, ntihomomorphism of a fuzzy and an anti-fuzzy group, Varahmihir Journal of mathematical Sciences, 4 (2)(2004) [9] N.Palaniappan, S. Naganathan, & K. rjunan, Study on Intuitionistic L-Fuzzy Subgroups, pplied mathematical Sciences, 3 (53) (2009) [10] Pandiammal. P, Natarajan. R, and Palaniappan. N, nti L-fuzzy M-subgroups, ntarctica J. Math., Vol. 7, number 6, , (2010). [11] Pandiammal. P, Intutitionistic anti L-fuzzy M- subgroups, International Journal of computer and organization rends Vol. 5 February(2014) [12] L.. Zadeh, Fuzzy sets, Information and control, 8, (1965) CONCLUSION In this paper, we define a new algebraic structure of Intuitionistic anti L-fuzzy translation and Homomorphism and anti homomorphism of Intuitionistic anti L- fuzzy ranslation, we wish to define Level subset of Intuitionistic anti L-fuzzy ranslation and other some L-fuzzy ranslation are in progress. ISSN: Page 18