A Study on Intuitionistic Multi-Anti Fuzzy Subgroups

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1 A Study on Intuitionistic Multi-Anti Fuzzy Subgroups R.Muthuraj 1, S.Balamurugan 2 1 PG and Research Department of Mathematics,H.H. The Rajah s College, Pudukkotta ,Tamilnadu, India. 2 Department of Mathematics,Velammal College of Engineering & Technology, Madurai ,Tamilnadu, India. ABSTRACT For any intuitionistic multi-fuzzy set A = { < x, µ A (x), ν A (x) > : x X} of an universe set X, we study the set [A] (α, β) called the (α, β) lower cut of A. It is the crisp multi-set { x X : µ i (x) α i, ν i (x) β i, i } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (α, β) lower cut sets. Keywords Intuitionistic fuzzy set (IFS), Intuitionistic multi-fuzzy set (IMFS), Intuitionistic multi-anti fuzzy subgroup (IMAFSG), Intuitionistic multi-anti fuzzy normal subgroup (IMAFNSG), (, ) lower cut, Homomorphism. Mathematics Subject Classification 20N25, 03E72, 08A72, 03F55, 06F35, 03G25, 08A05, 08A INTRODUCTION After the introduction of the concept of fuzzy set by Zadeh [14] several researches were conducted on the generalization of the notion of fuzzy set. The idea of Intuitionistic fuzzy set was given by Krassimir.T.Atanassov [1]. An Intuitionistic Fuzzy set is characterized by two functions expressing the degree of membership (belongingness) and the degree of non-membership (nonbelongingness) of elements of the universe to the IFS. Among the various notions of higher-order fuzzy sets, Intuitionistic Fuzzy sets proposed by Atanassov provide a flexible framework to explain uncertainity and vagueness. An element of a multi-fuzzy set can occur more than once with possibly the same or different membership values. In this paper we study Intuitionistic multi-anti fuzzy subgroup with the help of some properties of their (α, β) lower cut sets. This paper is an attempt to combine the two concepts: Intuitionistic Fuzzy sets and Multi-fuzzy sets together by introducing two new concepts called Intuitionistic Multi-fuzzy sets and Intuitionistic Multi-Anti fuzzy subgroups. 35

2 2. PRELIMINARIES In this section, we site the fundamental definitions that will be used in the sequel. 2.1 Definition [14] Let X be a non-empty set. Then a fuzzy set µ : X [0,1]. 2.2 Definition [9] Let X be a non-empty set. A multi-fuzzy set A of X is defined as A = { < x, µ A (x) > : x X } where µ A = ( µ 1, µ 2,, µ k ), that is, µ A (x) = ( µ 1 (x), µ 2 (x),, µ k (x) ) and µ i : X [0,1], i=1,2,,k. Here k is the finite dimension of A. Also note that, for all i, µ i (x) is a decreasingly ordered sequence of elements. That is, µ 1 (x) µ 2 (x) µ k (x), x X. 2.3 Definition [1] Let X be a non-empty set. An Intuitionistic Fuzzy Set (IFS) A of X is an object of the form A = {< x, µ(x), ν(x) > : x X}, where µ : X [0, 1] and ν : X [0, 1] define the degree of membership and the degree of non-membership of the element x X respectively with 0 µ(x) + ν(x) 1, x X. 2.4 Remark [1] (i) (ii) Every fuzzy set A on a non-empty set X is obviously an intuitionistic fuzzy set having the form A = {< x, µ(x), 1 µ(x) > : x X}. In the definition 2.3, When µ(x) + ν(x) = 1, that is, when ν(x) = 1 µ(x) = µ c (x), A is called fuzzy set. 2.5 Definition [13] Let A = {< x, µ A (x), ν A (x) > : x X } where µ A (x) = ( µ 1 (x), µ 2 (x),, µ k (x) ) and ν A (x) = ( ν 1 (x), ν 2 (x),, ν k (x) ) such that 0 µ i (x) + ν i (x) 1, for all i, x X. Here, µ 1 (x) µ 2 (x) µ k (x), x X. That is, µ i 's are decreasingly ordered sequence. That is, 0 µ i (x) + ν i (x) 1, x X, for i=1, 2,, k. Then the set A is said to be an Intuitionistic Multi-Fuzzy Set (IMFS) with dimension k of X. 2.6 Remark [13] Note that since we arrange the membership sequence in decreasing order, the corresponding non-membership sequence may not be in decreasing or increasing order. 2.7 Definition [13] Let A = { < x, µ A (x), ν A (x) > : x X } and B = { < x, µ B (x), ν B (x) > : x X } be any two IMFS s having the same dimension k of X. Then (i) A B if and only if µ A (x) µ B (x) and ν A (x) ν B (x) for all x X. 36

3 (ii) (iii) A = B if and only if µ A (x) = µ B (x) and ν A (x) = ν B (x) for all x X. A = { < x, ν A (x), µ A (x) > : x X } (iv) A B = { < x, (µ A B )(x), (ν A B )(x) > : x X }, where (µ A B )(x) = min{ µ A (x), µ B (x) } = ( min{µ ia (x), µ ib (x)} ) k i=1 and (ν A B )(x) = max{ ν A (x), ν B (x) } = ( max{ν ia (x), ν ib (x)} ) k i=1 (v) A B = { < x, (µ A B )(x), (ν A B )(x) > : x X }, where (µ A B )(x) = max{ µ A (x), µ B (x) } = ( max{µ ia (x), µ ib (x)} ) k i=1 and (ν A B )(x) = min{ ν A (x), ν B (x) } = ( min{ν ia (x), ν ib (x)} ) k i=1 Here { µ ia (x), µ ib (x) } represents the corresponding i th position membership values of A and B respectively. Also { ν ia (x), ν ib (x) } represents the corresponding i th position non-membership values of A and B respectively. 2.8 Theorem [13] For any three IMFS s A, B and C, we have : 1. Commutative Law A B = B A A B = B A 2. Idempotent Law A A=A A A=A 3. De Morgan's Laws (A B) = ( A B) (A B) = ( A B) 4. Associative Law A (B C) = (A B) C A (B C) = (A B) C 5. Distributive Law A (B C) = (A B) (A C) A (B C) = (A B) (A C) 2.9 Definition Let A = { < x, µ A (x), ν A (x) > : x X } be an IMFS and let α = (α 1,α 2,,α k ) [0,1] k and β = (β 1,β 2,, β k ) [0,1] k, where each α i, β i [0,1] with 0 α i + β i 1, i. Then (α, β)-lower cut of A is the set of all x such that µ i (x) α i with the corresponding ν i (x) β i, i and is denoted by [A] (α, β). Clearly it is a crisp multi-set Definition Let A = { < x, µ A (x), ν A (x) > : x X } be an IMFS and let α = (α 1,α 2,,α k ) [0,1] k and β = 37

4 (β 1,β 2,, β k ) [0,1] k, where each α i, β i [0,1] with 0 α i + β i 1, i. Then strong (α, β)-lower cut of A is the set of all x such that µ i (x) < α i with the corresponding ν i (x) > β i, i and is denoted by [A] (α, β)*. Clearly it is also a crisp multi-set. The following Theorem is an immediate consequence of the above definitions Theorem [13] Let A and B are any two IMFS s of dimension k drawn from a set X. Then A B if and only if [B] (α,β) [A] (α,β) for every α, β [0,1] k with 0 α i + β i 1 for all i Definition An intuitionistic multi-fuzzy set ( In short IMFS) A = { < x, µ A (x), ν A (x) > : x G } of a group G is said to be an intuitionistic multi-anti fuzzy subgroup of G ( In short IMAFSG ) if it satisfies the following : For all x,y G, (i) (ii) (iii) (iv) µ A (xy) max {µ A (x), µ A (y)} µ A (x -1 ) = µ A (x) ν A (xy) min {ν A (x), ν A (y)} ν A (x -1 ) = ν A (x) 2.13 Definition An intuitionistic multi-fuzzy set (In short IMFS) A = { < x, µ A (x), ν A (x) > : x G } of a group G is said to be an intuitionistic multi-anti fuzzy subgroup of G (In short IMAFSG) if it satisfies : (i) µ A (xy -1 ) max{µ A (x), µ A (y)} and (ii) ν A (xy -1 ) min{ν A (x), ν A (y)}, x,y G Remark (i) If A is an IFS of a group G, then we can not say about the complement of A, because it is not an IFS of G. (ii) If A is an IAFSG of a group G, then A c is need not be an IFS of G. (iii) A is an IMAFSG of a group G each IFS { < x, µ ia (x), ν ia (x) : x G > } k i=1 is an IAFSG of G. (iv) If A is an IMAFSG of a group G, then in general, we can not say A c is an IMFSG of the group G Definition An IMAFSG A = { < x, µ A (x), ν A (x) > : x G } of a group G is said to be an intuitionistic multianti fuzzy normal subgroup ( In short IMAFNSG ) of G if it satisfies : (i) µ A (xy) = µ A (yx) and (ii) ν A (xy) = ν A (yx), for all x,y G 38

5 2.15 Theorem An intuitionistic multi-anti fuzzy subgroup (IMAFSG) A of a group G is said to be normal if it satisfies : (i) µ A (g -1 xg) = µ A (x) and (ii) ν A (g -1 xg) = ν A (x), for all x A and g G Proof Let x A and g G be any element. Then µ A (g -1 xg) = µ A (g -1 (xg)) = µ A ((xg)g -1 ), since A is normal. = µ A (x(gg -1 )) = µ A (xe) = µ A (x). Hence the proof (i). Now, ν A (g -1 xg) = ν A (g -1 (xg)) = ν A ((xg)g -1 ), since A is normal. = ν A (x(gg -1 )) = ν A (xe) = ν A (x). Hence the proof (ii) Definition Let (G,.) be a groupoid and A, B be any two IMFS s having same dimension k of G. Then the product of A and B is denoted by A B and it is defined as : x G, A B(x) = ( µ A B (x), ν A B (x) ) where µ A B (x) = max [min{µ A (y), µ B (z)}: yz=x, y,z G] 0 k =(0, 0,, k times), if x is not expressible as x=yz and min [max{ν A (y), ν B (z)}:yz=x, y,z G] ν A B (x) = 1 k =(1, 1,, k times), if x is not expressible as x=yz That is, x G, ( max[min{µ A (y), µ B (z)}:yz=x, y,z G], min[max{ν A (y), ν B (z)}:yz=x, y,z G] ) A B(x) = (0 k,1 k ), if x is not expressible as x=yz That is, x G, (max[min{µ ia (y),µ ib (z)}:yz=x, y,z G],min[max{ν ia (y),ν ib (z)}:yz=x, y,z G]) k i=1 A B(x) = (0,1) k,if x is not expressible as x=yz where (0,1) k =((0,1), (0,1),, k times) 39

6 2.17 Definition Let X and Y be any two non-empty sets and f : X Y be a mapping. Let A and B be any two IMFS s having same dimension k, of X and Y respectively. Then the image of A( X) under the map f is denoted by f(a) and it is defined as : y Y, f(a)(y) = ( µ f(a) (y), ν f(a) (y) ) where µ f(a) (y) = max{µ A (x) : x f -1 (y)} 0 k, otherwise and ν f(a) (y) = min{ν A (x) : x f -1 (y)} 1 k, otherwise That is, f(a)(y) = ( max{µ ia (x) : x f -1 (y)}, min{ν ia (x) : x f -1 (y)} ) k i=1 (0,1) k, otherwise where (0,1) k =( (0,1), (0,1),, k times ) Also, the pre-image of B( Y) under the map f is denoted by f -1 (B) and it is defined as : x X, f -1 (B)(x) = ( µ B (f(x)), ν B (f(x)) ) 3. PROPERTIES OF (α, β) LOWER CUT OF INTUITIONISTIC MULTI-FUZZY SET In this section we shall prove some theorems on intuitionistic multi-anti fuzzy subgroups of a group G with the help of their (α, β) lower cuts. 3.1 Proposition If A and B are any two IMFS s of a universal set X, then their (α, β) lower cuts satisfies the following : (i) [A] (α, β) [A] (δ, θ) if α δ and β θ (ii) A B implies [B] (α, β) [A] (α, β) (iii) [A B] (α, β) = [A] (α, β) [B] (α, β) (iv) [A B] (α, β) [A] (α, β) [B] (α, β) (Here equality holds if α i + β i = 1, i ) (v) [ A i ] (α, β) = [A i ] (α, β),where α, β, δ, θ [0,1] k 3.2 Proposition Let (G,.) be a groupoid and A, B be any two IMFS s of G. Then we have [A B] (α, β) = [A] (α, β) [B] (α, β) where α, β [0,1] k. 3.3 Theorem 40

7 If A is an intuitionistic multi-anti fuzzy subgroup of a group G and α, β [0,1] k, then the (α, β)- lower cut of A, [A] (α, β) is a subgroup of G, where µ A (e) α, ν A (e) β and e is the identity element of G. Proof since µ A (e) α and ν A (e) β, e [A] (α, β). Therefore, [A] (α, β) φ. Let x,y [A] (α, β). Then µ A (x) α, ν A (x) β and µ A (y) α, ν A (y) β. Then i, µ ia (x) α i, ν ia (x) β i and µ ia (y) α i, ν ia (y) β i. max{µ ia (x), µ ia (y)} α i and min{ν ia (x), ν ia (y)} β i, i (1) µ ia (xy -1 ) max{µ ia (x), µ ia (y)} α i and ν ia (xy -1 ) min{ν ia (x), ν ia (y)} β i, i, since A is an intuitionistic multi-anti fuzzy subgroup of a group G and by (1). µ ia (xy -1 ) α i and ν ia (xy -1 ) β i, i. µ A (xy -1 ) α and ν A (xy -1 ) β xy -1 [A] (α, β) [A] (α, β) is a subgroup of G. Hence the Theorem. 3.4 Theorem The IMFS A is an intuitionistic multi-anti fuzzy subgroup of a group G each (α,β)-lower cut [A] (α, β) is a subgroup of G, α, β [0, 1] k. Proof From the above Theorem 3.3, it is clear. 3.5 Theorem If A is an intuitionistic multi-anti fuzzy normal subgroup of a group G and α, β [0, 1] k, then (α,β)-lower cut [A] (α, β) is a normal subgroup of G, where µ A (e) α, ν A (e) β and e is the identity element of G. Proof Let x [A] (α, β) and g G. Then µ A (x) α and ν A (x) β. That is, µ ia (x) α i and ν ia (x) β i i.(1) Since A is an intuitionistic multi-anti fuzzy normal subgroup of G, Hence the Theorem. 3.6 Theorem µ ia (g -1 xg) = µ ia (x) and ν ia (g -1 xg) = ν ia (x), i. µ ia (g -1 xg) = µ ia (x) α i and ν ia (g -1 xg) = ν ia (x) β i, i, by using(1). µ ia (g -1 xg) α i and ν ia (g -1 xg) β i, i. µ A (g -1 xg) α and ν A (g -1 xg) β g -1 xg [A] (α, β) [A] (α, β) is a normal subgroup of G. If A is an intuitionistic multi-fuzzy subset of a group G, then A is an intuitionistic multi-anti fuzzy subgroup of G each (α, β)-lower cut [A] (α, β) is a subgroup of G,for all α, β [0,1] k with α i + β i 1, i. 41

8 Proof Let A be an intuitionistic multi-anti fuzzy subgroup of a group G. Then by the Theorem 3.4, each (α, β)-lower cut [A] (α, β) is a subgroup of G for all α, β [0,1] k with α i + β i 1, i. Conversely, let A be an intuitionistic multi-fuzzy subset of a group G such that each (α, β)-lower cut [A] (α, β) is a subgroup of G for all α, β [0,1] k with α i + β i 1, i. To prove that A is an intuitionistic multi-anti fuzzy subgroup of G, we prove : (i) µ A (xy) max{µ A (x), µ A (y)} and ν A (xy) min{ν A (x), ν A (y)} for all x,y G (ii) µ A (x -1 ) = µ A (x) and ν A (x -1 ) = ν A (x) For proof (i): Let x,y G and for all i, let α i = max{µ ia (x), µ ia (y)} and β i = min{ν ia (x), ν ia (y)}. Then i, we have µ ia (x) α i, µ ia (y) α i and ν ia (x) β i, ν ia (y) β i That is, i, we have µ ia (x) α i, ν ia (x) β i and µ ia (y) α i, ν ia (y) β i Then we have µ A (x) α, ν A (x) β and µ A (y) α, ν A (y) β That is, x [A] (α, β) and y [A] (α, β) Therefore, xy [A] (α, β), since each (α, β)-lower cut [A] (α, β) is a subgroup by hypothesis. Therefore, i, we have µ ia (xy) α i = max{µ ia (x), µ ia (y)} and ν ia (xy) β i = min{ν ia (x), ν ia (y)}. That is, µ A (xy) max{µ A (x), µ A (y)} and ν A (xy) min{ν A (x), ν A (y)} and hence (i). For proof (ii): Let x G and i, let µ ia (x) = α i and ν ia (x) = β i. Then µ ia (x) α i and ν ia (x) β i is true i. Therefore, µ A (x) α and ν A (x) β Therefore, x [A] (α, β). Since each (α, β)-lower cut [A] (α, β) is a subgroup of G for all α, β [0,1] k and x [A] (α, β),we have x -1 [A] (α, β) which implies that µ ia (x -1 ) α i and ν ia (x -1 ) β i is true i. µ ia (x -1 ) µ ia (x) and ν ia (x -1 ) ν ia (x) is true i. Thus, i, µ ia (x)= µ ia ((x -1 ) -1 ) µ ia (x -1 ) µ ia (x) which implies that µ ia (x -1 ) = µ ia (x), i and hence µ A (x -1 ) = µ A (x). 42

9 And i, ν ia (x)= ν ia ((x -1 ) -1 ) ν ia (x -1 ) ν ia (x) which implies that ν ia (x -1 ) = ν ia (x), i and hence ν A (x -1 ) = ν A (x). Hence A is an intuitionistic multi-anti fuzzy subgroup of G and hence the Theorem. 3.7 Theorem If A and B are any two intuitionistic multi-anti fuzzy subgroups (IMAFSG s) of a group G, then (A B) is an intuitionistic multi-anti fuzzy subgroup of G. Proof since A and B are IMAFSG s of G, we have x,y G, (i) (ii) µ A (xy -1 ) max{µ A (x), µ A (y)} and ν A (xy -1 ) min{ν A (x), ν A (y)} µ B (xy -1 ) max{µ B (x), µ B (y)} and ν B (xy -1 ) min{ν B (x), ν B (y)}.(1) Now A B = { < x, µ A B (x), ν A B (x) > : x G } where µ A B (x) = max{ µ A (x), µ B (x) } and ν A B (x) = min { ν A (x), ν B (x) }. Then µ A B (xy -1 ) = max{ µ A (xy -1 ), µ B (xy -1 ) } max{ max{µ A (x), µ A (y)}, max{µ B (x), µ B (y)} }, by using (1) = max{ max{µ A (x), µ B (x)}, max{µ A (y), µ B (y)} } = max{ µ A B (x), µ A B (y) } and ν A B (xy -1 ) = min{ ν A (xy -1 ), ν B (xy -1 ) } min{ min{ν A (x), ν A (y)}, min{ν B (x), ν B (y)} }, by using (1) = min{ min{ν A (x), ν B (x)}, min{ν A (y), ν B (y)} } = min{ ν A B (x), ν A B (y) } That is, µ A B (xy -1 ) max{ µ A B (x), µ A B (y) } and ν A B (xy -1 ) min{ ν A B (x), ν A B (y) }, x,y G. Hence (A B) is an intuitionistic multi-anti fuzzy subgroup of G. Hence the Theorem. 3.8 Theorem The intersection of any two IMAFSG s of a group G need not be an IMAFSG of G. Proof Consider the abelian group G = { e, a, b, ab } with usual multiplication such that a 2 = e = b 2 and ab = ba. Let A = { < e, (0.2, 0.2), (0.7, 0.8) >, < a, (0.5, 0.5), (0.4, 0.4) >, < b, (0.5, 0.5), (0.2, 0.4) >, < ab, (0.4, 0.5), (0.2, 0.4) > } and B = { < e, (0.3, 0.1), (0.7, 0.8) >, < a, (0.8, 0.4), (0.2, 0.6) >, < b, (0.6, 0.4), (0.4, 0.5) >, < ab, (0.8, 0.4), (0.2, 0.5) > } be two IMFS s having dimension two of the group G. Clearly A and B are IMAFSG s of G. Then A B = { < e, (0.2, 0.1), (0.7, 0.8) >, < a, (0.5, 0.4), (0.4, 0.6) >, < b, (0.5, 0.4), (0.4, 0.5) >, < ab, (0.4, 0.4), (0.2, 0.5) > }. Here, it is easily verify that A B is not an IMAFSG of G. Hence the Theorem. 43

10 3.9 Theorem Let A and B be an IMAFSG s of a group G. But it is an uncertain to verify that A B is an IMAFSG of G. Proof This proof is done by the following two examples that are discussed in two cases : case(i) and case(ii). Case (i) : A and B are IMAFSG s of a group G A B is an IMAFSG of G but A B is not an IMAFSG of the group G. Consider the abelian group G = { e, a, b, ab } with usual multiplication such that a 2 = e = b 2 and ab = ba. Let A = { < e, (0.2, 0.2), (0.7, 0.8) >, < a, (0.5, 0.5), (0.4, 0.4) >, < b, (0.5, 0.5), (0.2, 0.4) >, < ab, (0.4, 0.5), (0.2, 0.4) > } and B = { < e, (0.3, 0.1), (0.7, 0.8) >, < a, (0.8, 0.4), (0.2, 0.6) >, < b, (0.6, 0.4), (0.4, 0.5) >, < ab, (0.8, 0.4), (0.2, 0.5) > } be two IMFS s having dimension two of the group G. Clearly A and B are IMAFSG s of G. Then A B = { < e, (0.3, 0.2), (0.7, 0.8) >, < a, (0.8, 0.5), (0.2, 0.4) >, < b, (0.6, 0.5), (0.2, 0.4) >, < ab, (0.8, 0.5), (0.2, 0.4) > } and A B = { < e, (0.2, 0.1), (0.7, 0.8) >, < a, (0.5, 0.4), (0.4, 0.6) >, < b, (0.5, 0.4), (0.4, 0.5) >, < ab, (0.4, 0.4), (0.2, 0.5) > }. Here, it is easily verify that A B is an IMAFSG of G but A B is not an IMAFSG of G. Hence case (i). Case (ii) : A and B are IMAFSG s of a group G both A B and A B are IMAFSG s of the group G. Consider the abelian group G = { e, a, b, ab } with usual multiplication such that a 2 = e = b 2 and ab = ba. Let A = { < e, (0.2, 0.1), (0.8, 0.8) >, < a, (0.5, 0.4), (0.4, 0.6) >, < b, (0.5, 0.4), (0.4, 0.5) >, < ab, (0.5, 0.4), (0.4, 0.5) > } and B = { < e, (0.3, 0.2), (0.7, 0.7) >, < a, (0.8, 0.5), (0.2, 0.4) >, < b, (0.6, 0.5), (0.4, 0.2) >, < ab, (0.8, 0.4), (0.2, 0.2) > } be two IMFS s having dimension two of the group G. Clearly A and B are IMAFSG s of G. Then A B = { < e, (0.3, 0.2), (0.7, 0.7) >, < a, (0.8, 0.5), (0.2, 0.4) >, < b, (0.6, 0.5), (0.4, 0.2) >, < ab, (0.8, 0.4), (0.2, 0.2) > } and A B = { < e, (0.2, 0.1), (0.8, 0.8) >, < a, (0.5, 0.4), (0.4, 0.6) >, < b, (0.5, 0.4), (0.4, 0.5) >, < ab, (0.5, 0.4), (0.4, 0.5) > }. Here, it is easily to verify that both A B and A B are IMAFSG s of G. Hence case (ii). From case (i) and case (ii), clearly it is an uncertain to verify that A B is an IMAFSG of G. Hence the Theorem Theorem Let A and B be any two IMAFSG s of a group G. Then A B is an IMAFSG of G A B=B A Proof Since A and B are IMAFSG s of G, each (α, β)-lower cuts [A] (α, β) and [B] (α, β) are subgroups of G, α,β [0,1] k with α i + β i 1, i (1) Suppose A B is an IMAFSG of G. 44

11 each (α, β)-lower cuts [A B] (α, β) are subgroups of G, α,β [0,1] k with α i + β i 1, i. Now, from (1), [A] (α, β) [B] (α, β) is a subgroup of G [A] (α, β) [B] (α, β) = [B] (α, β) [A] (α, β),since if H and K are any two subgroups of G, then HK is a subgroup of G HK=KH. Hence the Theorem Theorem [A B] (α, β) = [B A] (α, β), α,β [0,1] k with α i + β i 1, i. A B = B A If A is any IMAFSG of a group G, then A A = A. Proof Since A is an IMAFSG of a group G, each (α, β)-lower cut [A] (α, β) is a subgroup of G, α,β [0,1] k with α i + β i 1, i. [A] (α, β) [A] (α, β) = [A] (α, β), since H is a subgroup of G HH = H. [A A] (α, β) = [A] (α, β), α,β [0,1] k with α i + β i 1, i. A A = A Hence the Theorem. 4. INTUITIONISTIC MULTI-FUZZY COSETS In this section we shall prove some theorems on intuitionistic multi-fuzzy cosets of a group G. 4.1 Definition Let G be a group and A be an IMAFSG of G. Let x G be a fixed element. Then the set xa = {( g, µ xa (g), ν xa (g) ) : g G } where µ xa (g) = µ A (x -1 g) and ν xa (g) = ν A (x -1 g), g G is called the intuitionistic multi-fuzzy left coset of G determined by A and x. Similarly, the set Ax = { ( g, µ Ax (g), ν Ax (g) ) : g G } where µ Ax (g) = µ A (gx -1 ) and ν Ax (g) = ν A (gx -1 ), g G is called the intuitionistic multi-fuzzy right coset of G determined by A and x. 4.2 Remark It is clear that if A is an intuitionistic multi-anti fuzzy normal subgroup of G, then the intuitionistic multi-fuzzy left coset and the intuitionistic multi-fuzzy right coset of A on G coincides and in this case, we simply call it as intuitionistic multi-fuzzy coset. 4.3 Example Let G be a group. Then A = { < x, µ A (x), ν A (x) >, x G / µ A (x) = µ A (e) and ν A (x) = ν A (e) } is an 45

12 intuitionistic multi-anti fuzzy normal subgroup of G. Proof It is easy to verify. 4.4 Theorem Let A be an intuitionistic multi-anti fuzzy subgroup of a group G and x be any fixed element of G. Then the following are holds : (i) x[a] (α, β) = [xa] (α, β) (ii) [A] (α, β) x = [Ax] (α, β), α, β [0,1] k with α i + β i 1, i. Proof For proof (i), Now [xa] (α, β) = { g G : µ xa (g) α and ν xa (g) β } with α i + β i 1, i. Also x[a] (α, β) = x{ y G : µ A (y) α and ν A (y) β } = { xy G : µ A (y) α and ν A (y) β } (1) put xy = g y = x -1 g. Then (1) becomes as, x[a] (α, β) = { g G : µ A (x -1 g) α and ν A (x -1 g) β } = { g G : µ xa (g) α and ν xa (g) β } = [xa] (α, β) Therefore, x[a] (α, β) = [xa] (α, β), α, β [0,1] k with α i + β i 1, i. Hence the proof (i). For proof (ii), Now [Ax] (α, β) = { g G : µ Ax (g) α and ν Ax (g) β } with α i + β i 1, i. Also [A] (α, β) x = { y G : µ A (y) α and ν A (y) β }x = { yx G : µ A (y) α and ν A (y) β } (2) put yx = g y = gx -1. Then (2) becomes as, [A] (α, β) x = { g G : µ A (gx -1 ) α and ν A (gx -1 ) β } = { g G : µ Ax (g) α and ν Ax (g) β } = [Ax] (α, β) Therefore, [A] (α, β) x = [Ax] (α, β), α, β [0,1] k with α i + β i 1, i. Hence the proof (ii) and hence the Theorem. 4.5 Theorem Let A be an intuitionistic multi-anti fuzzy subgroup of a group G. Let x,y be any two elements of G such that α = max{ µ A (x), µ A (y) } and β = min{ ν A (x), ν A (y) }. Then the 46

13 following are holds : (i) xa = ya x -1 y [A] (α, β) (ii) Ax = Ay xy -1 [A] (α, β) (iii) Proof For (i), Now xa = ya [xa] (α, β) = [ya] (α, β), α, β [0,1] k with α i + β i 1, i. x[a] (α, β) = y[a] (α, β),by Theorem 4.4(i). x -1 y [A] (α, β),since each (α, β)-lower cut [A] (α, β) is a subgroup of G. Hence the proof (i). For (ii), Now Ax = Ay [Ax] (α, β) = [Ay] (α, β), α, β [0,1] k with α i + β i 1, i. [A] (α, β) x = [A] (α, β) y,by Theorem 4.4(ii). xy -1 [A] (α, β),since each (α, β)-lower cut [A] (α, β) is a subgroup of G. Hence the proof (ii) and hence the Theorem. 5. HOMOMORPHISM OF INTUITIONISTIC MULTI-ANTI FUZZY SUBGROUPS In this section we shall prove some theorems on intuitionistic multi-anti fuzzy subgroups of a group with the help of a homomorphism. 5.1 Proposition Let f : X Y be an onto map. If A and B are intuitionistic multi-fuzzy sets having the dimension k of X and Y respectively, then for each (α, β)-lower cuts [A] (α, β) and [B] (α, β), the following are holds : (i) f( [A] (α, β) ) [f(a)] (α, β) (ii) f -1 ( [B] (α, β) ) = [f -1 (B)] (α, β), α, β [0,1] k with α i + β i 1, i. Proof For (i), Let y f( [A] (α, β) ). Then there exists an element x [A] (α, β) such that f(x) = y. Then we have µ A (x) α and ν A (x) β,since x [A] (α, β). µ ia (x) α i and ν ia (x) β i, i. min{µ ia (x) : x f -1 (y)} α i and max{ν ia (x) : x f -1 (y) } β i, i. min{µ A (x) : x f -1 (y)} α and max{ν A (x) : x f -1 (y) } β µ f(a) (y) α and ν f(a) (y) β y [f(a)] (α, β) Therefore, f( [A] (α, β) ) [f(a)] (α, β), A IMFS(X). Hence the proof (i). For the proof (ii), Let x [f -1 (B)] (α, β) {x X : µ f -1 (B) (x) α, ν f -1 (B) (x) β } {x X : µ if -1 (B) (x) α i, ν if -1 (B) (x) β i }, i. {x X : µ ib (f(x)) α i, ν ib (f(x)) β i }, i. 47

14 Hence the proof (ii). 5.2 Theorem {x X : µ B (f(x)) α, ν B (f(x)) β } { x X : f(x) [B] (α, β) } { x X : x f -1 ( [B] (α, β) ) } f -1 ( [B] (α, β) ) Let f : G 1 G 2 be an onto homomorphism and if A is an IMAFSG of group G 1, then f(a) is an IMAFSG of group G 2. Proof By Theorem 3.6, it is enough to prove that each (α, β)-lower cuts [f(a)] (α, β) is a subgroup of G 2 for all α, β [0,1] k with α i + β i 1, i. Let y 1, y 2 [f(a)] (α, β). Then µ f(a) (y 1 ) α, ν f(a) (y 1 ) β and µ f(a) (y 2 ) α, ν f(a) (y 2 ) β µ if(a) (y 1 ) α i, ν if(a) (y 1 ) β i and µ if(a) (y 2 ) α i, ν if(a) (y 2 ) β i, i...(1) By the proposition 5.1(i), we have f( [A] (α, β) ) [f(a)] (α, β), A IMFS(G 1 ) Since f is onto, there exists some x 1 and x 2 in G 1 such that f(x 1 ) = y 1 and f(x 2 ) = y 2. Therefore, (1) becomes as, µ if(a) ( f(x 1 ) ) α i, ν if(a) ( f(x 1 ) ) β i and µ if(a) ( f(x 2 ) ) α i, ν if(a) ( f(x 2 ) ) β i, i. µ ia (x 1 ) µ if(a) ( f(x 1 ) ) α i, ν ia (x 1 ) ν if(a) ( f(x 1 ) ) β i and µ ia (x 2 ) µ if(a) ( f(x 2 ) ) α i, ν ia (x 2 ) ν if(a) ( f(x 2 ) ) β i, i. µ ia (x 1 ) α i, ν ia (x 1 ) β i and µ ia (x 2 ) α i, ν ia (x 2 ) β i, i. µ A (x 1 ) α, ν A (x 1 ) β and µ A (x 2 ) α, ν A (x 2 ) β max{µ A (x 1 ), µ A (x 2 )} α and min{ν A (x 1 ), ν A (x 2 )} β µ A (x 1 x -1 2 ) max{µ A (x 1 ), µ A (x 2 )} α A IMAFSG(G 1 ). and ν A (x 1 x 2-1 ) min{ν A (x 1 ), ν A (x 2 )} β,since µ A (x 1 x 2-1 ) α and ν A (x 1 x 2-1 ) β x 1 x 2-1 [A] (α, β) f(x 1 x 2-1 ) f( [A] (α, β) ) [f(a)] (α, β) f(x 1 )f(x 2-1 ) [f(a)] (α, β) f(x 1 )f(x 2 ) -1 [f(a)] (α, β) y 1 y 2-1 [f(a)] (α, β) [f(a)] (α, β) is a subgroup of G 2, α, β [0,1] k. 48

15 f(a) IMAFSG(G 2 ) Hence the Theorem. 5.3 Corollary If f : G 1 G 2 be a homomorphism of a group G 1 onto a group G 2 and { A i : i I } be a family of IMAFSG s of group G 1, then f( A i ) is an IMAFSG of group G Theorem Let f : G 1 G 2 be a homomorphism of a group G 1 into a group G 2. If B is an IMAFSG of G 2, then f -1 (B) is also an IMAFSG of G 1. Proof By Theorem 3.6, it is enough to prove that each (α, β)-lower cuts [f -1 (B)] (α, β) is a subgroup of G 1, α, β [0,1] k with α i + β i 1, i. Let x 1, x 2 [f -1 (B)] (α, β). Then it implies that µ f -1 (B) (x 1 ) α, ν f -1 (B) (x 1 ) β and µ f -1 (B) (x 2 ) α, ν f -1 (B) (x 2 ) β. µ B (f(x 1 )) α, ν B (f(x 1 )) β and µ B (f(x 2 )) α, ν B (f(x 2 )) β. max{ µ B (f(x 1 )), µ B (f(x 2 )) } α and min{ ν B (f(x 1 )), ν B (f(x 2 )) } β. µ B ( f(x 1 )f(x 2 ) -1 ) max{ µ B (f(x 1 )), µ B (f(x 2 )) } α and ν B ( f(x 1 )f(x 2 ) -1 ) min{ ν B (f(x 1 )), ν B (f(x 2 )) } β,since B IMAFSG(G 2 ). f(x 1 )f(x 2 ) -1 [B] (α, β) f(x 1 x 2-1 ) [B] (α, β),since f is a homomorphism. x 1 x 2-1 f -1 ( [B] (α, β) ) = [f -1 (B)] (α, β),by the proposition 5.1(ii). x 1 x 2-1 [f -1 (B)] (α, β) [f -1 (B)] (α, β) is a subgroup of G 1 f -1 (B) is an IMAFSG of G 1. Hence the Theorem. 5.5 Theorem Let f : G 1 G 2 be a surjective homomorphism and if A is an IMAFNSG of group G 1, then f(a) is also an IMAFNSG of group G 2. 49

16 Proof Let g 2 G 2 and y f(a). Since f is surjective, there exists g 1 G 1 and x A such that f(x) = y and f(g 1 ) = g 2. Since A is an IMAFNSG of G 1, µ A (g -1 1 xg 1 ) = µ A (x) and ν A (g -1 1 xg 1 ) = ν A (x), x A and g 1 G 1. Now consider, µ f(a) (g -1 2 yg 2 ) = µ f(a) ( f(g -1 1 xg 1 ) ),since f is a homomorphism. Similarly, we can easily prove that ν f(a) (g 2-1 yg 2 ) = ν f(a) (y). Hence f(a) is an IMAFNSG of G 2 and hence the Theorem. 5.6 Theorem = µ f(a) (y ),where y = f(g 1-1 xg 1 ) = g 2-1 yg 2 = min{ µ A (x ) : f(x ) = y for x G 1 } = min{ µ A (x ) : f(x ) = f(g 1-1 xg 1 ) for x G 1 } = min{ µ A (g 1-1 xg 1 ) : f(g 1-1 xg 1 )=y = g 2-1 yg 2 for x A, g 1 G 1 } = min{ µ A (x) : f(g 1-1 xg 1 ) = g 2-1 yg 2 for x A, g 1 G 1 } = min{ µ A (x) : f(g 1 ) -1 f(x)f(g 1 ) = g 2-1 yg 2 for x A, g 1 G 1 } = min{ µ A (x) : g 2-1 f(x)g 2 = g 2-1 yg 2 for x G 1 } = min{ µ A (x) : f(x) = y for x G 1 } = µ f(a) (y) If A is an IMAFNSG of a group G, then there exists a natural homomorphism f : G G/A is defined by f(x) = xa, x G. Proof Let f : G G/A be defined by f(x) = xa, x G. Claim1: f is a homomorphism That is, to prove: f(xy) = f(x)f(y), x,y G. That is, to prove: (xy)a = (xa)(ya), x,y G. Since A is an IMAFNSG of G, we have µ A (g -1 xg) = µ A (x) and ν A (g -1 xg) = ν A (x), x A and g G. Or, equivalently, µ A (xy) = µ A (yx) and ν A (xy) = ν A (yx), x,y G. Also, g G, we have (xa)(g) = ( µ xa (g), ν xa (g) ) = ( µ A (x -1 g), ν A (x -1 g) ) 50

17 (ya)(g) = ( µ ya (g), ν ya (g) ) = ( µ A (y -1 g), ν A (y -1 g) ) [(xy)a](g) = ( µ (xy)a (g), ν (xy)a (g) ) = ( µ A [(xy) -1 g], ν A [(xy) -1 g] ) g G, by definition 2.16, we have [(xa)(ya)](g) = ( max[min{µ xa (r), µ ya (s)} : g = rs ], min[max{ν xa (r), ν ya (s)} : g = rs ] ) = ( max[min{µ A (x -1 r), µ A (y -1 s)} : g = rs], min[max{ν A (x -1 r), ν A (y -1 s)} : g = rs ] ) Claim2: µ A [(xy) -1 g] = max[ min{µ A (x -1 r), µ A (y -1 s)} : g = rs ] and ν A [(xy) -1 g] = min[ max{ν A (x -1 r), ν A (y -1 s)} : g = rs ], g G. Now consider µ A [(xy) -1 g] = µ A [y -1 x -1 g] = µ A [y -1 x -1 rs],since g = rs. = µ A [y -1 (x -1 rsy -1 )y] = µ A [x -1 rsy -1 ],since A is normal. max{ µ A (x -1 r), µ A (sy -1 ) },since A is IMAFSG of G. = max{ µ A (x -1 r), µ A (y -1 s) }, g = rs G, since A is normal Therefore, µ A [(xy) -1 g] = min[ max{ µ A (x -1 r), µ A (y -1 s) } : g = rs ], g G. = max[ min{ µ A (x -1 r), µ A (y -1 s) } : g = rs ], g G. Similarly,we can easily prove ν A [(xy) -1 g] = min[ max{ ν A (x -1 r), ν A (y -1 s) } : g = rs ], g G. Hence the claim2. Thus, [(xy)a](g) = [(xa)(ya)](g), g G. (xy)a = (xa)(ya) f(xy) = f(x)f(y) f is a homomorphism Hence the claim1 and hence the Theorem. 6. CONCLUSION In the theory of fuzzy sets, the level subsets are vital role for its development. Similarly, the (α, β)-lower cut of an intuitionistic multi-fuzzy sets are very important role for the development of the theory of intuitionistic multi-fuzzy sets. In this paper an attempt has been made to study some algebraic natures of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (α, β) lower cut sets. REFERENCES [1] Atanassov K.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), no.1, [2] Basnet D.K. and Sarma N.K., A note on Intuitionistic Fuzzy Equivalence Relation, International Mathematical Forum, 5, 2010, no.67, [3] Biswas R., Vague Groups, International Journal of Computational Cognition, Vol.4, no.2, June [4] Das P.S., Fuzzy groups and level subgroups, Journal of Mathematical Analysis and Applications, 84 (1981), [5] Kul Hur and Su Youn Jang, The lattice of Intuitionistic fuzzy congruences, International Mathematical Forum, 1, 2006, no.5, [6] Mukharjee N.P. and Bhattacharya P., Fuzzy normal subgroups and fuzzy cosets, Information Sciences, 34 (1984),

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