Commutative Multi Anti L – Fuzzy Subgroups

Commutative Multi Anti L – Fuzzy Subgroups

K. Sunderrajan, M. Suresh

Department of Mathematics,

SRMV College of Arts and Science, Coimbatore-641020, Tamilnadu, India.

R. Muthuraj

Department of Mathematics,

H.H.The Rajahs College, Pudukkottai622 001, Tamilnadu, India

Abstract

In this paper, we define the algebraic structures of multi-anti fuzzy subgroup and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and group theory in multi-anti fuzzy subgroups. In this paper the concepts of L Fuzzy subgroup and anti L Fuzzy subgroups and some results involving them are generalized to the L Fuzzy case where L is an arbitrary Lattice with 0 and 1. Commutativity for Multi Anti L Fuzzy subgroup is introduced and some necessary and sufficient conditions for a Multi Anti L Fuzzy subgroup to be commutative are derived.

Keywords

L fuzzy subgroup, Multi L Fuzzy subgroups, Anti L Fuzzy subgroup, Multi Anti L Fuzzy subgroup, Normal L Fuzzy subgroup, Commutative L Fuzzy subgroup, Commutative Multi Anti L Fuzzy subgroup.

1. Introduction

   Applying the concept of Fuzzy sets introduced by Zadeh [11] to group theory Rosenfeld [7] defined Fuzzy subgroup of a given group and derived some of their properties.

   Das [4] characterized fuzzy subgroups by their level subgroups. The concept of anti fuzzy subgroup was introduced by Biswas [3]. The Concept of Commutativity L-fuzzy subgroups was introduced by Souriar Sebastian and S.Babu Sundar [10]. In all these studies, the closed unit interval [0, 1] is taken as the Membership lattice.

   In this paper, we extend these concepts to the Multi Anti L Fuzzy [7 ] case, where L is an arbitrary Lattice and derive some more properties. We also introduce commutative for Multi Anti L Fuzzy subgroups and obtain some characterizations.

2. Preliminaries

Throughout this paper G denotes an arbitrary Multiplicative group with e is an identity element and L denotes an arbitrary Lattice with least element 0 and greatest element

1. The join and meet operations in L are denoted by and respectively. A function A: G

   L is called and Multi L Fuzzy subset of G. If H G then H denotes the characteristic function of H.

   1. Definition [11]

      Let X be any nonempty set. A fuzzy set of X is : X [0, 1].

   2. Definition [11]

   A L-Fuzzy subset A of X is a mapping from X into L, where L is a complete lattice satisfying the infinite meet distributive law. if L is the unit interval [0,1] of real numbers, there are the usual fuzzy subset of X.

   A L-fuzzy subset A: XL is said to be non-empty, if it is not the constant map which assumes the values 0 of L.

2. 3 Definition [5]

   A L-fuzzy subset A of G is said to be a L-fuzzy group of G, if for all x,y G

   1. A(xy ) A(x) A(y)

   2. A(x-1) = A (x) .

1. 4 Definition [3]

   A L-fuzzy subset A of G is said to be a anti L-fuzzy group of G, if for all x,y G

   1. A(xy ) A(x) A(y)

   2. A(x-1) = A (x) .

2. Some properties of multi Anti L- fuzzy subgroups

   In this section, we discuss some of the properties of multi Anti L- fuzzy

   subgroups.

   1. Definition [9]

      Let X be a non empty set. A Multi L fuzzy set A in X is defined as a set of ordered sequences. A = { (x, 1(x), 2(x), …, i(x), …) : x X}, where i : X L for all i.

   2. Definition [5]

      A Multi L Fuzzy subset A of G is called an Multi L Fuzzy subgroup (MLFS) of G if for every x,y G,

      Remark

      1. A(xy ) A(x) A(y)

      2. A(x-1) = A (x)

      It can be proved that if A is an MLFS of G then A(e) A(x) for all x,y G. Also a L-Fuzzy subset A of G is an MLFS of G iff A(xy1) A(x) A(y) for all x, y G.

   3. Definition [10]

      A Multi anti L Fuzzy subset A of G is called an Multi anti L Fuzzy subgroup (MALFS) of G if for every x,y G,

      1. A(xy ) A(x) A(y)

      2. A(x-1) = A (x) .

   Remark

   It can be proved that if A is an MALFS of G then A(e) A(x) for all x,y G. Also a L-Fuzzy subset Aof G is an MALFS of G iff A(xy1) A(x) A(y) for all x, y

   G.

   1. Theorem

      1. For any non-empty subset H of G, the characteristic function of H, H is an MLFS of G iff H is a subgroup of G

      2. If A is an MALFS of G then A(e) A(x) for every x G.

      3. Multi L Fuzzy subset A is an MALFS of G iff A(x y1) A(X) A(Y) for every x,y G.

      4. An Multi L Fuzzy subset A of G is an MLFS of G iff

   Aa = { x G ; A(x) a } is a subgroup of G, for every aL for which

   0 < a A (e). The subgroup Aa is called the level subgroup of A determined by a.

   Proof

   Their proofs are straight forward.

   1. Definition [10]

      If c: L L is an order reversing involution satisfying De Morgan Law and Ac denotes c(a) for every a L, then for every aL then for any Multi Anti L Fuzzy subset A of G, Ac : G L defined by Ac (x) = (A(x))c for every xG is called the c- Complement of A.

   2. Definition[8]

   Let A= (1, 2, …, k) be a multi-fuzzy set of dimension k and let i be the fuzzy complement of the ordinary fuzzy set i for i = 1, 2, …, k. The Multi-fuzzy

   Complement of the multi-fuzzy set A is a multi-fuzzy set (1, …, k) and it is denoted by c(A) or A' or Ac.

   That is, c(A) = {(x, c(1(x)), …, c(k(x) )) : x X} = {(x, 1 1(x), …, 1 k(x)

   ) : x X}, where c is the fuzzy complement operation.

   3.2 Theorem

   A is a Multi L Fuzzy subgroup of G iff Ac is a Multi anti L Fuzzy subgroup of G.

   Proof

   Suppose A is a multi-fuzzy subgroup of G. Then for all x, y G , A (xy) min {A (x ), A (y)}

   1 Ac(xy) min { (1 Ac(x)),(1 Ac(y))}

   Ac(xy) 1 min { (1 Ac(x)),(1 Ac(y))}

   Ac (xy) max { Ac(x), Ac(y)}. We have, A(x) = A(x1) for all x in G

   1 Ac(x) = 1 Ac(x1)

   Therefore Ac(x) = Ac(x1) .

   Hence A c is a multi-anti L- fuzzy subgroup of G.

   Corollary

   If L has an order reversing involution satisfying De Morgan Law and H is a non empty subset of G. Then H is an Multi anti L Fuzzy subgroup (MALFS) of G iff H is the set complement of a subgroup of G.

   Proof

   H is an MALFS of G.

   cH =G-H is an MLFS of G.

   G H is a subgroup of G.

   1. Definition [9]

      Let X be a non empty set. A Multi L Fuzzy set A in X is defined as a set of ordered sequences. A = { (x, 1(x), 2(x), …, i(x), …) : x X}, where i : X [0, 1] for all i.

      Remark

      1. If the sequences of the membership functions have only k-terms (finite number of terms), k is called the dimension of A.

      2. The set of all multi-fuzzy sets in X of dimension k is denoted by FS(X).

      3. The multi fuzzy membership function A is a function from X to such that for all x in X, A(x) = (1(x), 2(x), …, k(x)).

      4. For the sake of simplicity, we denote the multi-fuzzy set

         A = {(x, 1(x), 2(x), …, k(x) ) : x X} as A= (1, 2, …, k).

   2. Definition [9]

      Let k be a positive integer and let A and B in FS(X), where A= (1, 2, …,

      k) and B = (1, 2…, k),then we have the following relations and operations:

      1. A B if and only if i i, for all i = 1, 2, …, k;

      2. A = B if and only if i = i, for all i = 1, 2, …, k;

         iii. AB = ( 11, …, kk ) ={( x, max(1(x), 1(x)), …, max(k(x), k(x)) ) : x X};

         iv. AB = (11, …, kk) = {( x, min(1(x), 1(x)), …, min(k(x), k(x)) ) : x X};

         v. A + B = (1+1,. …, k+k) ={( x, 1(x) + 1(x) 1(x)1(x), …, k(x) + k(x) k(x)

         k(x) ) : x X}.

   3. Definition [8]

      Let A be a fuzzy set on a group G. Then A is said to be a fuzzy subgroup of G if for all x, y G,

      1. A(xy) min { A(x) , A(y)}

      2. A(x -1) = A(x).

   4. Definition [8]

      A multi-fuzzy set A of a group G is called a multi-fuzzy subgroup of G if for all x, y G ,

      1. A (xy ) min {A(x), A(y)}

      2. A(x-1) = A (x) .

   5. Definition [8]

      A multi-fuzzy set A of a group G is called a multi-anti fuzzy subgroup of G if for all x, y G,

      1. A(xy) max {A(x), A(y)}

      2. A(x-1) = A (x)

   6. Definition [8]

      Let A and B be any two multi-fuzzy sets of a non-empty set X. Then for all x X,

      1. A B iff A(x) B(x),

      2. A = B iff A(x) = B(x),

      3. AB (x) = max {A(x), B(x)},

      4. AB (x) = min {A(x), B(x)}.

   7. Definition [8]

      Let A and B be any two multi-fuzzy sets of a non-empty set X. Then

      1. AA = A, AA = A,

      2. A AB, B AB, AB A and AB B,

      3. A B iff AB = B,

      4. A B iff AB = A.

   1. Theorem

      Let A be a multi-anti fuzzy subgroup of a group G and e is the identity element of

      1. Then

         Proof

         1. A(x) A(e) for all x G .

         2. The subset H = {xG / A(x) = A(e)} is a subgroup of G.

            1. Let xG .

               A (x) = max { A (x) , A (x) }

               = max { A (x) , A (x-1) }

               A (xx-1)

               = A (e).

               Therefore, A (x) A (e), for all xG.

            2. Let H = {xG / A(x) = A(e)} Clearly H is non-empty as eH.

      Let x , y H. Then, A(x) = A(y) = A(e)

      A(xy-1) max {A(x), A(y-1)}

      = max {A(x), A(y)}

      = max {A(e), A(e)}

      = A(e)

      That is, A(xy-1) A(e) and obviously A(xy-1) A(e) by i. Hence, A(xy-1) = A(e) and xy-1 H.

      Clearly, H is a subgroup of G.

   2. Theorem

      Let A be any multi-anti fuzzy subgroup of a group G with identity e. Then A(xy1) = A(e) A(x) = A(y) for all x ,y in G.

      Proof

      Given A is a multi-anti fuzzy subgroup of G and A (xy1) = A(e) .

      Then for all x , y in G ,

      A(x) = A(x(y 1y))

      = A((xy1)y)

      max { A(xy1), A(y)}

      = max { A(e) , A(y)}

      = A(y).

      That is, A(x) A(y).

      Now, A(y) = A(y1) , since A is a multi-anti fuzzy subgroup of G.

      = A(ey1)

      = A((x1x)y1)

      = A(x1(x y1))

      max { A(x1) , A(x y1)}

      = max {A(x) , A(e)}

      = A(x).

      That is, A(y) A(x).

      Hence, A(x) = A(y).

   3. Theorem

      A is a multi-anti fuzzy subgroup of a group G if and only A(x y1) max {A (x), A (y)}, for all x, y in G .

      Proof

      Let A be a multi-anti fuzzy subgroup of a group G. Then for all x ,y in G , A (x y) max {A (x), A (y)}

      and A (x) = A (x1).

      Now , A(x y1) max { A(x) , A( y1)}.

      = max { A(x) , A(y)}

      A(x y1) max { A(x ) , A( y)}.

   4. Theorem

      If A is an Multi L Fuzzy subset of G, then the following are equivalent.

      1. A is both an MLFS and MALFS of G.

      2. A is constant.

         Proof

         i ii

         If A is both an MLFS and MALFS G, then we have A (e) A(x) A (e) for every X G,

         Hence A (x) = A (e) for all xG and therefore, A is constant. The converse is trivial.

   5. Theorem

      If L is a chain and A is an MALFS of G then A (xy) = A(yx) = A(x) A(y) for every x, y G with A (x) A (y).

      Proof

      Since L is a chain A(x) A(y) without loss of generality. We assume that A(x) < A(y). Then,

      A(xy ) A(x) A(y)

      = A(x)

      = A(xyy-1)

      A(xy) A(y) Since A(x) < A(y). we have A(xy) = A(xy) A(y).

      Thus, A(xy) A(x) A(xy) and hence by the anti circularity law for lattices, A (xy) = A(x).

      In a similar way, we can prove that A(yx) = A(x). Hence, A(xy)= A (yx) = A(x) = A(x) A(y).

   6. Theorem

      If L is A chain and A is an MALFS of G, then the following are equivalent.

      1. A(xy) = A(x) A(y) whenever A(x) = A(y),

      2. A is a constant.

   Proof

   In view of theorem 3.7, Implies that A(xy) = A(x) A(y) for every x, y G.

   Putting y = x-1 we get A(x) = A(e) for every x G.Hence i ii.

   The converse is obvious.

   Remark

   Theorem 3.7 – 3.8 imply that if L is a Chain then the equality in the first axiom of the definitions of MLFS and MALFS hold only for constants.

   3.13 Definition [10]

   A lattice L is said to be without zero meets a b > 0 for every a,bL such that a > 0 and b > 0.

   Every chain is a lattice without zero meets. If A is an Multi Anti L Fuzzy subset of G, then the support of A is defined as the set supp (A) = { x G;A(x)>0}.

   3.9 Theorem

   Let L be a lattice without zero meets If A(

   0 ) is an MALFS of G then supp (A) is a

   subgroup of G. Further, all subgroups of G can be realized as the support of some MALFS of G.

   Proof

   meets

   Let x,y supp (A). Then A(x) > 0 and A(y) > 0. Since L is a lattice without zero

   A(xy-1) > A(x) A(y) > 0.Hence xy-1 supp (A) and therefore supp (A) is a subgroup of G. Now let H be any subgroup of G. Fix a L such that a > 0. Define A : G L by

   A(x) =

   a if xH

   0 otherwise

   Then A is an MALFS of G and supp (A) = H.

   3.1 Example

   Let G = {1,-1,i,-i} where i =

   1 . This is a group under usual multiplication of

   complex numbers. Let = [0,1]. Then L is a lattice without zero meets. Define A:G L by A(1) = ½, A(-1) = 1 A(i) = A(-i) = 0. Then A is an multi L-Fuzzy subset of G and supp

   (A) = {1,-1} is a subgroup of G. But A is not an MLFS of G, since A(-1,-1) = A(1) = ½ and A(-1) A(-1) = 1 and hence A(-1-1) < (A(-1) A(-1).

3. Properties of Commutative Multi Anti L Fuzzy subgroups

Throughout this section we assume that L is an Lattice withput zero meets. If A( ) is an MALFS of G , then the restriction of A to supp(A). we shall denote A supp (A) also by A.

4.1Definition [10]

An MALFS A of G is said to be commutative if Axy = A yx for every x,yG with A(x)

> 0 and A(y) > 0. Where Axy denotes the restriction of A (considered as a function) to the singletion subset {xy} of G.

It may be noted that commutativity of A requires xy and yx to coincide whenever x,y supp (A). Observe that this definition actually generalizes the notion of commutativity of ordinary subgroups. That is, for any non-empty subset H of G, H is a commutative MALFS of G iff H is a commutative subgroup of G.

1. Theorem

   Let A( 0 ) be an MALFS of G. Then the following are equivalent.

   (i)A is a commutative MALFS of G

   1. Supp (A) is a commutative subgroup of G.

   2. The level subgroups Aa are commutative subgroups of G, for every a L with 0 < a A(e).

      Proof

      1. (ii)

         Let A be a commutative MALFS of G. By theorem(3.9), supp (A) is a subgroup of G. Let x,y supp (A). Since A is commutative. Axy = A yx xy = yx.

      2. (iii)

         Assume that supp (A) is a commutative subgroup of G. Let aL such that 0 < a A(e). By theorem (3.1) (d) Aa is a subgroup of G. Let x,yAa.

         Then A(x) a and A(y) a.Since a > 0, we have x,ysupp (A) and hence xy = yx

      3. (i)

      Assume (iii) let x,y G such that A(x) > 0 and A(y) > 0. Let A(x) = a1 and A(y)= a2. Then x Aa1 and ya2. Put a = a1 a2. Since L is without zero meets, a > 0. Also a a1, a a2. So that Aa Aa1 and Aa Aa2. Therefore x,y Aa. But Aa is a commutative subgroup of

      G. Hence xy = yx and therefore Axy = Ayx.

2. Theorem:

If A is a commutative MALFS of G and supp (A) is a normal subgroup of G. Then A is a normal MALFS of G.

Proof:

Let x,y G. We have three different cases.

Case (i)

x,y supp (A). Then by Definition [4.1] Axy = Ayx. Hence A(xy) = A(yx).

Case (ii)

X supp (A) and y supp (A). Then both xy and yx does not belong to supp (A).

Hence A(xy) = A(yx) = 0. Case (iii)

x,y supp (A). Then xy and yx may or may not belong to supp (A). Since supp (A) is normal subgroup of G, either xy and yx both belongs to supp (A) or both does not belong to supp (A).

1. Xy, yx supp (A), then A(xy) = A(yx) =0

2. If xy, yx supp (A) then by Theorem (4.2) xy = yx and hence A(xy) = A(yx).

4.1Example

Let L = [0,1] and G be any non commutative group. Since G is a normal subgroup of itself, Gis normal MALFS of G, But supp(G)= G is not commutative. HenceG is not commutative MALFS of G.

1. Theorem

   For any group G the following are equivalent.

   1. G is commutative

   2. All MALFSs of G are commutative

   3. G is a commutative MALFS of G.

      Proof:

      (i)(ii)

      Let G be a commutative group and A be any MALFS of G. For any x,y G xy = yx and hence Axy = Ayx. Hence any MALFS of G is commutative.

      1. (iii)

         Trivial, since G itself is an MALFS of G.

      2. (i)

      Let G be a commutative MALFS of G.

      Then by theorem (4.1) supp (G) = G is commutative.

      n

      i

      i

      Let Gi (i = 1,2 . n) be groups. G= G

      i1

      be their product and i : G Gi

      be the projections defined byi(x1,x2,.xn) = xi The direct product of Multi L- Fuzzy

      n

      i

      i

      subsets Ai of Gi (i=1,2. n) is defined as the Multi Anti L Fuzzy subset of A = A of G

      i1

      given by

      A(x) = { Ai (i (x)) :i = 1,2,3,n }

2. Theorem

n

If Ai is a (commutative) MALFS of Gi for each i= 1,2 . n then Ai is a

i 1

n

i

i

(commutative) MALFS of G .

i1

Proof:

n n

Let A = Ai and G = Gi for x = ( x1, x2 . Xn), y = (y1, y2 . Yn) G.

i 1 i 1

We have

A(xy-1) = {Ai(xiyi-1) : i=1,2,n}

{Ai(xi ) Ai (yi-1) : i=1,2,n}

= (I Ai(xi ) ) (I Ai(yi-1))

= A(x) A(y)

Hence by theorem 3.1 (c) A is an MLFS of G.

Now let Ai be the commutative MALFS\s and x,ysupp (A). Then A(x) = { Ai (xi ) : i = 1,2,3n } >0.Ai (xi) > 0 i = 1,2,3..n.

Similarly Ai(yi) > 0 i= 1,2 n Hence xi , yi supp (Ai) i= 1,2 n. Since each Ai is

a commutative MALFS by Theorem 4.2 xi yi = yI xi for all i= 1,2 n. Hence xy = yx. Thus supp (A) is a commutative and hence A is a commutative MALFS of G.

The converse of the above proposition is not true.

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