International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com bstract. The (α,β cut o Intuitionistic uzz set has been discussed b the author in [5].Let : X Y be a mapping. In this paper, a relation between the intuitionistic uzz set o X and that o intuitionistic uzz set ( o Y has been obtained with the help o their (α,β cut sets. When is a homomorphism rom group G to the group G 2, some results concerning the intuitionistic uzz subgroups o G and G 2 has been obtained. Mathematics Subject Classiication : 03F55, 06D72, 0872 Kewords: Intuitionistic uzz set (IFS, Intuitionistic uzz subgroup (IFSG, Intuitionstic uzz normal subgroup (IFNSG, (α, β cut., Homomorphism. Introduction ter the introduction o the concept o uzz set b Zadeh [6] several researches were conducted on the generalization o the notion o uzz set. The idea o Intuitionistic uzz set was given b Krassimiri T. tanassov []. In this paper we stud some results relating to the Intuitionistic uzz subgroup o a group with its homomorphic image b using the properties o their (α, β cut sets. 2. Preliminaries Deinition (2.[] Let X be a ixed non-empt set. n Intuitionistic uzz set (IFS o X is an object o the ollowing orm = { < x, (x, ν (x > : x
370 P. K. Sharma X}, where :X [0,] and ν :X [0, ] deine the degree o membership and degree o non-membership o the element x X respectivel and or an x X, we have 0 (x + ν (x. Remark (2.2: When (x + ν (x =, i.e. when ν (x = - (x = c (x. Then is called uzz set. Deinition (2.3[] Let = { < x, (x, ν (x > : x X} and = { < x, (x, ν (x > : x X} be an two IFS s o X, then (i i and onl i (x (x and ν (x ν (x or all x X (ii = i and onl i (iii = { < x, ( (x, (ν ν (x (x = (x and ν (x = ν (x or all x X > : x X}, where ( (x = Min{ (x, (x} = (x (x and (ν ν (x = Max{ ν (x, ν (x } = ν (x ν (x (iv = { < x, ( (x, (ν ν (x > : x X}, where ( (x = Max{ (x, (x} = (x (x and (ν ν (x = Min{ ν (x, ν (x } = ν (x ν (x Deinition(2.4 [ 3 ] : Let ( X,. be a groupoid and, be two IFS s o X. Then the Intuitionistic uzz product o and is denoted b ο and is deined as ollows : For an x X ο( x = ( ( x, ν ( x [ ( ( z] z x (x 0 ; i x is not expressible as x= z = ο i.e. ( x, ο ο, where [ ν( ν(] z z x (x ; i x is not expressible as x= z = ν ο ( [ ( ( z], [ ν ( ν ( z] z x z x ο = = ( 0, ; i x is not expressible as x = z Deinition (2.5: Let be Intuitionistic uzz set o a universe set X. Then ( α, β -cut o is a crisp subset C α, β ( o the IFS is given b C α, β ( = { x : x X such that (x α, ν (x β }, where α, β [ 0, ] with α + β.
Homomorphism o intuitionistic uzz groups 37 Proposition (2.6[ 5 ]: I and be two IFS s o a universe set X, then ollowing holds (i C α, β ( C δ, θ( i α δ and β θ (ii (iii (iv C -β, β ( C α, β( C α, -α ( implies C α, β ( C α, β ( C α, β ( = C α, β ( C α, β ( (v C α, β ( C α, β ( C α, β ( equalit hold i α + β = (vi C α, β ( i = C α, β ( i (vii C 0, ( = X Proposition (2.7 : Let ( X,. be a groupoid and, be two IFS s o X. Then C α,β (ο = C α,β ( C α,β ( Proo. Now C α,β (ο = { g X : ο (g α, ν ο (g β } Let g C α,β (ο ο (g α, ν ο (g β [ ( (] z z= g ο (g = and [ ν( ν(] z z= g ν ο (g = 0 ; i x is not expressible as g = z ; i x is not expressible as g = z ( ( z] α and ν ( ν ( z] β [ [ z= g z= g s, z and 2, z 2 in X such that g = z and g = 2 z 2 and ( ( z α and ν ( ν ( z β 2 2 ( α, ( z α and ν ( β, ν ( z β 2 2 let ν ( - α, ν ( z - α and ( - β, ( z -β 2 2 C α,α(, z C α,α( and 2 C β, β(, z 2 C β, β( Proposition (2.6, we have C β, β( C α, β( C α,α( ; or ever IFS's o X g = C ( C ( C (C ( i.e. g C (C ( Hence 2z2 β, β β, β α, β α, β α, β α, β C α,β (ο = C α,β ( C α,β ( Proposition(2.8 Let and be two IFSG s o group G. Then ο is IFSG o group G i and onl i ο = ο
372 P. K. Sharma Proo. Now ο is IFSG o group G C α,β (ο is subgroup o group G or all α,β [0,] with α + β. s and be two IFSG s o group G C α,β ( and C α,β ( are subgroup o group G or all α, β [0,] with α + β. Now C α,β ( C α,β ( is subgroup o group G C α,β ( C α,β ( = C α,β (C α,β ( [ s or subgroups H and K o a group G, HK is subgroup o G i HK = KH ] C α,β (ο = C α,β (ο ; or all α,β [0,] with α + β. ο = ο Corollar (2.9 Let be IFSG o group G, then ο = Proo : Since is IFSG o group G C α,β ( is subgroup o group G, or all α,β [0,] with α + β C α,β ( C α,β ( = C α,β ( [ s H is subgroup o group G HH = H ] C α,β ( ο = C α,β (, or all α,β [0,] with α + β. ο = Deinition (2.0 [ 3 ]: Let X and Y be two non-empt sets and : X Y be a mapping. Let and be IFS s o X and Y respectivel. Then the image o under the map is denoted b ( and is deined as ( = (, ν (, where ( ( ( ( ( ( { ( x : x (} 0 ; otherwise and ν ( ( { ν ( x : x (} i.e. ( ( { ( x : x (}, { ν ( x : x (} ; otherwise ( ( 0, ; otherwise lso the pre-image o under is denoted b - ( and is deined as x = ( ( x, ν ( x ( ( ( ( where ( x = ( ( x and ν ( x = ν ( ( x i.e. ( ( ( ( x = ( ( (, ν ( ( x x Proposition (2. : Let : X Y be a mapping. Then the ollowing holds C ( C ( (, IFS( X (i ( α, β α, β (ii ( C ( = C ( (, IFS( Y α, β α, β
Homomorphism o intuitionistic uzz groups 373 Proo. ( i Let (C α, β( be an element, then s x C α, β( such that (x = and (x α and ν (x β { ( x : x (} α and { ν ( x : x (} β ie.. ( α and ν ( β i.e. C α, β( (, ( ( Hence ( α, β α, β αβ, ( ( ( ( C ( C ( (, IFS( X { α ν β } ( ii C = x X : ( x, ( x { x X α ν β } x X C ( = : ( (x, ( (x { αβ, } x X : x ( Cαβ, ( ( Cαβ, ( = : (x = = { } 3. Homomorphism o Intuitionistic uzz groups Theorem(3. : Let : G G 2 be surjective homomorphism and be IFSG o group G. Then ( is IFSG o group G 2. Proo. Now b Theorem 3.5 o [ 5 ], it is enough to show that C α, β( ( is subgroup o G 2 or all α, β [ 0, ] with α + β. Let, 2 C α, β( ( be an two elements, then ( ( α, ν ( ( β and ( ( 2 α, ν ( ( 2 β Prop. (2.(i we have ( Cα, β( Cα, β( (, IFS( G Thereore s x and x 2 in G such that ( x ( α, ν ( x ν ( β and ( x ( α, ν ( x ν ( β ( ( ( ( ( 2 2 2 2 ( x α, ν( x β and ( x2 α, ν( x2 β ( x ( x2 α and ν( x ν( x2 β s is IFSG o group G. Thereore ( xx ( x ( x α and ν ( xx ν ( x ν ( x β x x C ( x x C ( C ( ( 2 2 2 2 ( xx 2 α and ν ( xx 2 β 2 α, β ( 2 ( α, β α, β ( x ( x2 Cα, β ( ( 2 Cα, β ( ( Hence C ( ( is a subgroup o G α, β 2
374 P. K. Sharma Corollar (3.2: I :G G 2 be homomorphism o group G onto a group G 2 and { j : j I } be a amil o IFSG s o group G, then ( j is IFSG o group G 2. Theorem (3.3 : Let : G G 2 be homomorphism o group G into a group G 2. Let be IFSG o group G 2. Then - ( is IFSG o group G.. Proo. Theorem 3.5 o [ 5 ], it is enough to show that C α, β( - ( is subgroup o G or all α, β [ 0, ] with α + β. Let x, x 2 C α, β( - ( be an two elements, then ( x α, ν ( x β and ( x α, ν ( x β ( ( ( ( 2 2 ie.. ( ( x α, ν ( ( x β and ( ( x α, ν ( ( x β 2 2 ( ( x ( ( x2 α and ν( ( x ν( ( x2 β s is IFSG o group G 2. Thereore ( ( x ( x ( ( x ( ( x α ν ( ( x ( x ν ( ( x ν ( ( x β 2 2 2 2 ( ( x ( x α and ν ( ( x ( x β 2 2 ( x ( x C ( ( xx C ( 2 αβ, 2 αβ, x x2 ( Cαβ, = Cαβ, xx C ( ( 2 αβ, ( ( ( [ Prop. (2.(ii ] Hence C ( ( is a subgroup o group G αβ, Theorem(3.4 : Let : G G 2 be surjective homomorphism and be IFNSG o group G. Then ( is IFNSG o group G 2. Proo : Let g 2 G 2 and ( be an elements. Then s g G and x such that (x = and (g = g 2. s is IFNSG o group G Thereore ( g - x g = ( x and ν ( g - x g = ν ( x ; x and g G Now ( ( g 2 - g 2 = ( ( ( g - x g [ s is homomorphism ] = ( (, where = ( g - x g = g 2 - g 2 = { ( x : (x = or x G } = { ( x : (x = ( g - x g or x G } = { (g - x g :(g - x g = = g 2 - g 2 or x, g G } = { ( x : ( g - x g = g 2 - g 2 or x, g G } = { ( x : ( g - (x (g = g 2 - g 2 or x, g G }
Homomorphism o intuitionistic uzz groups 375 = { ( x : g - 2 (x g 2 = g - 2 g 2 or x G } = { ( x : (x = or x G } = ( ( Similarl, we can show that ν ( ( g - 2 g 2 = ν ( ( Hence ( is IFNSG o group G 2. Theorem(3.4 : Let G be a group and be IFNSG o group G. Then there exists a natural homomorphism : G G/ deined b (x = x ; or all x G Proo. Let : G G/ be a mapping deined b (x = x ; or all x G We show that is homomorphism i.e. (x = (x ( ; or all x, G i.e. (x = (x( ; or all x, G Since is IFNSG o group G, thereore we have ( g - x g = ( x and ν ( g - x g = ν ( x ; x and g G Or equivalentl, ( x = ( x and ν ( x = ν ( x ; x, G x g ( g, ν ( g ( x = = g, ν ( x g ; g G lso ( ( ( x x ( ( ( ( g ν g ( g ν g ( ( ( ( x g ν( x g ( x g ν x g g = (, ( = (, ( ; g G (x g = (, ( = ((, (( ; g G ( ( ( x νx ν g= rs g= rs = ( ( x r ( s, ν( x r ν( s g= rs g= rs Now (x( g = ( r ( s, ( r ( s ; g G We claim that (( x g = ( x r ( s and ν(( x g ν( x r ν( s ; g G g= rs = g= rs Now (( x g = ( x g = ( x rs = ( ( x rs = ( x rs ( x r ( s = ( x r ( s ; g = rs G Thus (( x g = ( x r ( s ; g G g= rs
376 P. K. Sharma Similarl, we can show that ( ( ν ν ν (( x g = ( x r ( s ; g G g= rs Thus ( x ( g = ( x( ( g holds ; g G (x = (x ( ( x = ( x ( Hence is hom om orphism Theorem(3.5 : Let = { < x, (x, ν (x >, x G such that. (x = (e, ν (x = ν (e} be IFNSG o group G and be an IFS o group G and : G G/ be natural homomorphism deined b (x = x, or all x G, then - ( ( = ο Proo : Let x G be an element, then [ ] [ ] ( ( ( z (, ν ( z ν ( (o x = x= z x= z (0, otherwise ( [ ( ( ], [ ( ( ] z ν z ν z= x z= x (0, otherwise ( ( (, ( ( x ν x ν z= x z= x (0, otherwise ( [ ( ( ], [ ( ( ] e ν e ν z= x z= x (0, otherwise ( [ ( ], [ ( ] ν z = x z = x (0, o th erw ise ( [ ( ], [ ( ] ν = z x = z x (0, o th erw ise
Homomorphism o intuitionistic uzz groups 377 ( (, ( z x ν z x z x z x (0, otherw ise ( x = ( ( ( x, ν ( ( ( x = ( ( ( x, ν ( ( ( x lso ( ( ( ( [ ( ], [ ( ν ] ( x = ( ( x = ( (0, otherwise ( [ ( ], [ ν( ] = z x = z x, where z = x (0, otherwise ( (, ( z x ν z x = z x z x (0, otherwise Thus ( x = (o x, or all x G ( ( ( ( = ( Hence o ( ( [ ( ], [ ( ] ν x x (0, otherwise Reerences [] K.T tanassov, Intuitionistic uzz sets, Fuzz Sets and Sstems 20(986, no., 87-96 [2] D.K asnet and N.K Sarma, note on Intuitionistic Fuzz Equivalence Relation, International Mathematical Forum, 5, 200, no. 67, 330-3307 [3] Kul Hur and Su Youn Jang, The lattice o Intuitionistic uzz congruences", International Mathematical Fourum,, 2006, no. 5, 2-236 [4] N Palaniappan, S Naganathan and K rjunan, stud on Intuitionistic L-Fuzz Subgroups, pplied Mathematical Sciences, vol. 3, 2009, no. 53, 269-2624
378 P. K. Sharma [5] P.K. Sharma, ( α, β Cut o Intuitionistic uzz Groups International Mathematical Forum,Vol. 6, 20, no. 53, 2605-264 [6] L. Zadeh, Fuzz sets, Inormation and Control 8, (965, 338-353 Received: Ma, 20