The Homomorphism and Anti-Homomorphism of N-Generated Fuzzy Groups and Its Level Subgroups

Transcription

1 Inernaional Journal o Emerging Engineering Research and Technology Volume 3, Issue 8, Augus 205, PP 2-26 ISSN (Prin) & ISSN (Online) The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is Level Subgroups *Address or correspondence: mariamahs2545@gmail.com A.Solairaju, S.Rehinakumar 2, M.Maria Arockia Raj 3 P&Research Deparmen o Mahemaics, Jamal Mohamed College (Auonomous), Tiruchirappalli, Tamilnadu, India 2 P&Research Deparmen o Mahemaics, Thanhai Hans Roever College (Auonomous), Perembalur, Tamilnadu,India 3 Deparmen o Mahemaics, Mahalakshmi Engineering College, Tiruchirappalli, Tamilnadu, India ABSTRACT In his paper, we inroduce some properies o level subgroups o n-generaed uzzy subgroups o a group wih respec o he homomorphism and ani-homomorphism. Keywords: Fuzzy se, muli-uzzy se, uzzy subgroup, muli-uzzy subgroup, ani-uzzy subgroup, muli-ani uzzy subgroup, n generaed uzzy subse, n generaed uzzy subgroups, n generaed uzzy level subses, n-generaed uzzy level subgroups, homomorphism and ani-homomorphism INTRODUCTION Aer he inroducion o he concep o uzzy ses by L.A.Zadeh [], researchers were conduced he generalizaions o he noion o uzzy ses, A. Roseneld [2] inroduced he concep o uzzy group and he idea o inuiionisic uzzy se was irs published by K.T. Aanassov [3]. S. sabu sebasian and T.V Ramakrishnan [4,5] inroduced he concep o Muli-uzzy ses and Muliuzzy subgroups.also R.Muhuraj and S.Balamurugan [6] produced some resuls in Muli-uzzy groups and is lower level subgroups.choudhury.f.p. and Chakrabory.A.B. And Khare.S.S. [7] Deined a uzzy subgroup and uzzy homomorphism.in his chaper we inroduce some properies o n generaed uzzy subgroups o a group wih homomorphism and ani-homomorphism PRELIMINARIES Deiniion Le X be a non-empy se. A uzzy se A on X is a mapping A: X 0, and is deined as A x X / x, ( x ) Deiniion Le X and Y be any wo ses. Le : X Y be a uncion. I is a uzzy se on X hen he image o under is a uzzy se on Y deined by ( ) y v( y) Sup ( x), y Y is called image o under x ( y) Deiniion Le X and Y be any wo ses. Le : X Y b a uncion. I S is a uzzy se on Y hen he preimage o S under is a uzzy se on X & is deined by Deiniion (S) ( x) S ( x ). Le A be a uzzy subse o a se X. For [0, ], A x X / A( x) subse o A is called a level uzzy Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205 2

2 A.Solairaju e al. The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is Deiniion Le X be a non-empy se, J be an indexing se and L j : j J a amily o parially ordered ses. A muli-uzzy se A in X is a se X A x, j ( x) : x X, j L j, j J j J Deiniion Le X be a non-empy se and A be a muli-uzzy se on X. An n is o he orm generaed uzzy se on X X A x, ( x) : x X, L, i P, n N Where Deiniion i i k n 0 ( x) x X, n 0, ( x) i ( x) k i Le X and Y be any wo ses. Le : X Y be a uncion. I is a n generaed uzzy se on X hen he image o under is a n generaed uzzy se ony and is deined by ( ) y v( y) Sup ( x), y Y is called image o under x ( y) I S is an n generaed uzzy se on Y hen he preimage o X is deined by (S) ( x) S ( x ) Deiniion Le A be an n-generaed uzzy se on X. For [0, ], A x X / ( x) is called a n- generaed level uzzy subse o A Properies (). A B x ( x ) (2). A B x ( x ) (3). A B x ( x ) x, max x, ( x) ; x X (4). A B x ( x ) x, min x, ( x) ; x X (5). A B x, x ( x) x ( x) ; x X (6). I A x, x ; x X, hen A x, x ; x X Deiniion C Le be a group. A uzzy subse A o is said o be a uzzy subgroup o i ( i). A( xy) min A( x), A( y ) ( ii). A x A( x) x, y Deiniion: Le be a group. A n generaed uzzy subse o a group is called a n generaed 22 Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205

3 A.Solairaju e al. The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is uzzy subgroup o i ( i). ( xy) min ( x), ( y ) ( ii). x ( x) x, y k n k n where x i ( x), ( y) i ( y) k i k i Deiniion k n, & ( xy ) i ( xy k ) Le be a group. A n generaed uzzy subse o a group is called a n generaed ani- uzzy subgroup o ( i). ( xy) max ( x), ( y ) ( ii). x ( x) x, y Deiniion i Le, and ', be any wo groups, hen he uncion : ' is called a group homomorphism i xy ( x) ( y) x, y Deiniion Le, and ', be any wo groups, hen heuncion : ' is called a group ani- homomorphism i xy ( y) ( x) x, y Le and ' be any wo groups wih ideniy. Le : ' be a homomorphism hen (i). () ',where and are he ideniies o and ' respecively. i (ii). ( a ) ( a ) Proposiion Le A be a uzzy subgroup o a group.then or [0, ] suchha ( e), A is a subgroup o Proposiion The homomorphic image o a uzzy subgroup o a group is a uzzy subgroup o a group Proposiion The homomorphic pre- image o a uzzy subgroup o a group Proposiion ' ' is a uzzy subgroup o a group The ani- homomorphic image o a uzzy subgroup o a group is a uzzy subgroup o a group Proposiion The ani- homomorphic pre- image o a uzzy subgroup o a group ' ' is a uzzy subgroup o a group The homomorphic image o a level subgroup o a n-generaed uzzy subgroup o a group is a level subgroup o an n-generaed uzzy subgroup o a group Proo Le and ' be any wo groups ' 23 Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205

4 A.Solairaju e al. The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is Le : ' be a homomorphism Tha is xy ( x) ( y) x, y Le V, where an n- generaed uzzy subgroup o a group V is an n- generaed uzzy subgroup o Le x, y ( x) and ( y) in ' ' is a level subgroup o Tha is ( x) and ( y) and ( xy ) We have o prove ha is a level subgroup o V Now, V ( x) ( x) V ( x) V ( y) ( y) V ( y) And ( ) ( ) ( ) ( ), V x y V x y As is a hom V ( xy ), as is a hom ( xy ) V ( x) ( y) Is a level subgroup o an n-generaed uzzy subgroup o? ' The homomorphic pre- image o a level subgroup o a n-generaed uzzy subgroup o a group ' is a level subgroup o a n-generaed uzzy subgroup o a group Proo Le and ' be any wo groups Le : ' be a homomorphism Tha is xy ( x) ( y) x, y Le V, where an n- generaed uzzy subgroup o a group ' is an n- generaed uzzy subgroup o Le ( x ) and ( y) in ' x, y is a level subgroup o V Tha is V ( x) & V ( y), And V ( x) ( y) We have o prove ha is a level subgroup o Now, x V ( x) x 24 Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205

5 A.Solairaju e al. The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is y V ( y) y And xy V xy V ( x) ( y ) As is a homomorphism V ( x) ( y) Is a level subgroup o an n- generaed uzzy subgroup o a group? ' The ani- homomorphic image o a level subgroup o a n-generaed subgroup o a group is a level subgroup o an n-generaed uzzy subgroup o a group Proo Le and ' be any wo groups Le : ' be an ani- homomorphism Tha is xy ( y) ( x) x, y Le V, where an n- generaed uzzy subgroup o group ' V is an n- generaed uzzy subgroup o Le x, y ( x) and ( y) in ' ' is a level subgroup o Tha is ( x) and ( y) and ( y x) We have o prove ha is a level subgroup o V Now, V ( x) ( x) V ( x) V ( y) ( y) V ( y) And ( ) ( ) ( ) ( ), V y x V y x As is a ani- hom V ( y x ) As is a ani- hom ( y x) V ( y) ( x) Is a level subgroup o an n-generaed uzzy subgroup V o a group? ' The ani- homomorphic pre- image o a level subgroup o a n-generaed uzzy subgroup o group ' is a level subgroup o a n-generaed uzzy subgroup o a group a Proo Le and ' be any wo groups Le : ' be an ani- homomorphism Tha is xy ( y) ( x) x, y 25 Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205

6 A.Solairaju e al. The Homomorphism and Ani-Homomorphism o N-eneraed Fuzzy roups and Is Le V, where an n- generaed uzzy subgroup o is ' is an n- generaed uzzy subgroup o Le ( x ) and ( y) in ' x, y is a level subgroup o V Tha is V ( x) & V ( y), And V ( y) ( x) We have o prove ha is a level subgroup o x V ( x) x y V ( y) y And xy V xy V ( y ) ( x ) As is a homomorphism V ( y) ( x) xy Is a level subgroup o an n- generaed uzzy sub group REFERENCES o a group? [] L.A.Zadeh, Fuzzy ses, Inormaion and conrol, 8(965), [2] A.Roseneld, Fuz groups,j.mah.anal.appl.,35( [3] K.T.Aanassov Inuiionisic uzzy ses,fuzzy ses and Sysems 20(986) no., [4] S.Sabu and T.V. Ramakrishnan, Mui-uzzy ses, Inernaional Mahemaical Forum, 50(200), [5] S.Sabu and T.V. Ramakrishnan Muli-uzzy subgroups,inernaional journal o conemporary Mahemaical Science, 6(8) (20), [6] N.Palaniappan and R.Muhuraj, Ani-uzzy group and Is lower level subgroups, Anarica J.Mah., ()(2004), 7-76 [7] Choudhury.F.P. and Chakrabory.A.B. and Khare.S.S., A noe on uzzy subgroups and homomorphi.journal o mahemaical analysis and applicaions 3, (988) 26 Inernaional Journal o Emerging Engineering Research and Technology V3 I8 Augus 205