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 Authors : K. Sunderrajan, M. Suresh, R. Muthuraj
 Paper ID : IJERTV2IS4538
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 18042013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Commutative Multi Anti L – Fuzzy Subgroups
K. Sunderrajan, M. Suresh
Department of Mathematics,
SRMV College of Arts and Science, Coimbatore641020, 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 multianti fuzzy subgroup and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and group theory in multianti 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.

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 Lfuzzy 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.

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

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.

Definition [11]
Let X be any nonempty set. A fuzzy set of X is : X [0, 1].

Definition [11]
A LFuzzy 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 Lfuzzy subset A: XL is said to be nonempty, if it is not the constant map which assumes the values 0 of L.


3 Definition [5]
A Lfuzzy subset A of G is said to be a Lfuzzy group of G, if for all x,y G

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

A(x1) = A (x) .


4 Definition [3]
A Lfuzzy subset A of G is said to be a anti Lfuzzy group of G, if for all x,y G

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

A(x1) = A (x) .


Some properties of multi Anti L fuzzy subgroups
In this section, we discuss some of the properties of multi Anti L fuzzy
subgroups.

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.

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

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

A(x1) = 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 LFuzzy subset A of G is an MLFS of G iff A(xy1) A(x) A(y) for all x, y G.


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,

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

A(x1) = 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 LFuzzy subset Aof G is an MALFS of G iff A(xy1) A(x) A(y) for all x, y
G.

Theorem

For any nonempty subset H of G, the characteristic function of H, H is an MLFS of G iff H is a subgroup of G

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

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

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.

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.

Definition[8]
Let A= (1, 2, …, k) be a multifuzzy set of dimension k and let i be the fuzzy complement of the ordinary fuzzy set i for i = 1, 2, …, k. The Multifuzzy
Complement of the multifuzzy set A is a multifuzzy 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 multifuzzy 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 multianti 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 =GH is an MLFS of G.
G H is a subgroup of G.

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

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

The set of all multifuzzy sets in X of dimension k is denoted by FS(X).

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)).

For the sake of simplicity, we denote the multifuzzy set
A = {(x, 1(x), 2(x), …, k(x) ) : x X} as A= (1, 2, …, k).


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:

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

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}.


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,

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

A(x 1) = A(x).


Definition [8]
A multifuzzy set A of a group G is called a multifuzzy subgroup of G if for all x, y G ,

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

A(x1) = A (x) .


Definition [8]
A multifuzzy set A of a group G is called a multianti fuzzy subgroup of G if for all x, y G,

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

A(x1) = A (x)


Definition [8]
Let A and B be any two multifuzzy sets of a nonempty set X. Then for all x X,

A B iff A(x) B(x),

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

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

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


Definition [8]
Let A and B be any two multifuzzy sets of a nonempty set X. Then

AA = A, AA = A,

A AB, B AB, AB A and AB B,

A B iff AB = B,

A B iff AB = A.


Theorem
Let A be a multianti fuzzy subgroup of a group G and e is the identity element of

Then
Proof

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

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

Let xG .
A (x) = max { A (x) , A (x) }
= max { A (x) , A (x1) }
A (xx1)
= A (e).
Therefore, A (x) A (e), for all xG.

Let H = {xG / A(x) = A(e)} Clearly H is nonempty as eH.


Let x , y H. Then, A(x) = A(y) = A(e)
A(xy1) max {A(x), A(y1)}
= max {A(x), A(y)}
= max {A(e), A(e)}
= A(e)
That is, A(xy1) A(e) and obviously A(xy1) A(e) by i. Hence, A(xy1) = A(e) and xy1 H.
Clearly, H is a subgroup of G.


Theorem
Let A be any multianti 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 multianti 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 multianti 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).

Theorem
A is a multianti 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 multianti 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)}.

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

A is both an MLFS and MALFS of G.

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.


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(xyy1)
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).

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

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

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 = x1 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(xy1) > A(x) A(y) > 0.Hence xy1 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) = Â½, A(1) = 1 A(i) = A(i) = 0. Then A is an multi LFuzzy 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) = Â½ and A(1) A(1) = 1 and hence A(11) < (A(1) A(1).


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 nonempty subset H of G, H is a commutative MALFS of G iff H is a commutative subgroup of G.

Theorem
Let A( 0 ) be an MALFS of G. Then the following are equivalent.
(i)A is a commutative MALFS of G

Supp (A) is a commutative subgroup of G.

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

(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.

(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

(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.



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).

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

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.

Theorem
For any group G the following are equivalent.

G is commutative

All MALFSs of G are commutative

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.

(iii)
Trivial, since G itself is an MALFS of G.

(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 }



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(xy1) = {Ai(xiyi1) : i=1,2,n}
{Ai(xi ) Ai (yi1) : i=1,2,n}
= (I Ai(xi ) ) (I Ai(yi1))
= 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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