E212126

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org Volume 2 Issue 1 ǁ January. 2013 ǁ PP.21-26
www.ijesi.org 21 | P a g e
L-fuzzy sub  -group and its level sub  -groups
K.Sunderrajan, A.Senthilkumar, R.Muthuraj
Department of Mathematics, SRMV College of Arts and Science, Coimbatore-20.
Department of Mathematics, SNS college of Technology, Coimbatore-35
Department of Mathematics, H.H.The Rajah’s College, Pudukkottai–01.
ABSTRACT: In this paper, we discussed some properties of L-fuzzy sub  -group of a lattice ordered group
and define a new algebraic structure of an anti L-fuzzy sub  -group and some related properties are
investigated. We establish the relation between L-fuzzy sub  -group and anti L-fuzzy sub  -group. The
purpose of this study is to implement the fuzzy set theory and group theory in L-fuzzy sub  -group and anti L-
fuzzy sub  -groups. Characterizations of level subsets of a L-fuzzy sub  -group are given. We also discussed
the relation between a given a L-fuzzy sub  -group and its level sub  -groups and investigate the conditions
under which a given sub  -group has a properly inclusive chain of sub  -groups. In particular, we formulate
how to structure an L-fuzzy sub  -group by a given chain of sub  -groups.
Keywords–– Fuzzy set, Fuzzy sub group, L-fuzzy sub l-group of a group, Anti L-Fuzzy Sub l-Group of a
group. AMS Subject Classification (2000): 20N25, 03E72, 03F055, 06F35, 03G25.
I. INTRODUCTION
L. A. Zadeh [14] introduced the notion of a fuzzy subset A of a set X as a function from X into [0,
1]. Rosenfeld [8] applied this concept in group theory and semi group theory, and developed the theory of
fuzzy subgroups and fuzzy sub semi groupoids respectively. J.A. Goguen [2] replaced the valuations set [0,
1], by means of a complete lattice in an attempt to make a generalized study of fuzzy set theory by studying
L-fuzzy sets. In fact it seems in order to obtain a complete analogy of crisp mathematics in terms of
fuzzy mathematics, it is necessary to replace the valuation set by a system having more rich algebraic
structure. These concepts  -groups play a major role in mathematics and fuzzy mathematics. Satya
Saibaba[13] introduced the concept of L- fuzzy  -group and L-fuzzy  -ideal of  -group.
We wish to define a new algebraic structure of L-fuzzy sub  -group and establishes the relation with
L-fuzzy sub  - group and discussed some of its properties.
II. PRELIMINARIES
In this section we site the fundamental definitions that we will be used in the sequel. Throughout this
paper ,*)(GG  is a group, e is the identity element of G and xy we mean x*y.
2.1 Definition [13]
A lattice ordered group or a  -group is a system G = (G,  , ≤), where
i. ( G, ) is a group,
ii. ( G, ≤ ) is a lattice ,
iii. The inclusion is invariant under all translations bxax  i.e. byabxayx 
Remark
Throughout this paper G = (G,  , ≤) is a lattice ordered group or a  -group, e is the identity element of G and
xy we mean xy.
2.2 Definition [8]
Let S be any non-empty set. A fuzzy subset  of S is a function : S  [ 0 ,1 ].
2.3 Definition [8]
Let G be a group. A fuzzy subset  of G is called a fuzzy subgroup if for any x , y  G,
i. (xy) ≥ (x)  (y),
ii. (x-1
) = (x).

2.4 Definition [1]
Let G be a group. A fuzzy subset  of G is called an anti fuzzy subgroup if for any x , y  G,
i. (xy) ≤ (x)  (y),
ii. (x-1
) = (x).
2.5 Definition [13]
An L the infinite meets distributive law. If L is the unit interval [0, 1] of real numbers, there are the usual fuzzy
subsets of G. A L fuzzy subset : G  Lis said to be non-empty, if it is not the constant map which assumes the
values 0 of L.
2.6 Definition [11]
A L-fuzzy subset  of G is said to be a L-fuzzy subgroup of G if for any x , y  G,
i. (xy) ≥ (x)  (y),
ii. (x-1
) = (x).
2.7 Definition [11]
A L-fuzzy subset  of G is said to be an anti L-fuzzy subgroup of G if for any x , y  G,
i. (xy) ≤ (x)  (y),
ii. (x-1
) = (x).
2.8 Definition[13]
A L-fuzzy subset  of G is said to be a L-fuzzy sub l group of G if for any x , y  G,
i. (xy) ≥ (x)  (y),
ii. (x-1
) = (x),
iii. (xy) ≥ (x)  (y),
iv. (xy) ≥ (x)  (y).
2.9 Definition [13]
A L-fuzzy subset  of G is said to be an anti L-fuzzy sub l group of G if for any x , y  G,
i. (xy) ≤ (x)  (y),
ii. (x-1
) = (x),
iii. (xy) ≤ (x)  (y),
iv. (xy) ≤ (x)  (y).
III. PROPERTIES OF L-FUZZY SUB l -GROUP OF G
In this section we discuss some of the properties of L-fuzzy sub l -group of G.
3.1 Theorem
Let  be a L-fuzzy sub  -group of G then ,
i. (x) ≤  (e) for x  G, where e is the identity element of the G.
ii. The Subset H = { x  G / (x) = (e) } is a sub  -group of G.
Proof
i. Let x  G, then
(x) = (x)  (x) ,
(x) = (x)  (x-1
),
≤ (xx-1
),
= (e).
That is, (x) ≤ (e).
ii. Let H = { x  G / (x) = (e) }.
Clearly H is non-empty as e  H and for any x , y  G, we have,
(x) = (y) = (e).
Now, (xy-1
) ≥ (x)  (y-1
),
= (x)  (y),
= (e)  (e),
That is , (xy-1
) ≥ (e) and obviously (xy-1
) ≤ (e), by ( i ).
Hence, (xy-1
) = (e) and xy-1
 H.
Hence , H is a sub  -group of a group G.

3.2 Theorem
Let  be a L-fuzzy sub  -group of G with identity e , then, for any x , y  G, (xy-1
) = (e)  (x) = (y).
Proof
Let  be a L-fuzzy sub  -group of G with identity e and (xy-1
) = (e), then for for any x , y  G, we have,
(x) = (x(y-1
y))
= ((xy-1
) y)
≥ ( xy-1
)  (y)
= (e)  (y)
= (y).
That is, (x) ≥ (y).
Now, (y) = (y-1
), then
= (ey-1
)
=  ((x-1
x)y-1
)
= (x-1
(xy-1
))
≥ (x-1
)  (xy-1
)
≥ (x)  (e)
(y) ≥ (x).
Hence, (x) = (y).
3.3 Theorem
Let  be a L-fuzzy sub  -group of G iff (xy-1
) ≥ (x)  (y) for any x , y  G.
Proof
Let  be a L-fuzzy sub  -group of G, then for any x , y  G , we have
(xy) ≥ (x)  (y),
Now, (xy-1
) ≥ (x)  (y-1
),
= (x)  (y),
 (xy-1
) ≥ (x)  (y).
3.4 Theorem
Let  and  be any two L-fuzzy sub  -group of G, then  is a L-fuzzy sub  -group of G.
Proof
Let  and  be an L-fuzzy sub l -group of G.
i. ( )(xy) = (xy)   (xy)
≥ ((x)  (y))  ((x)  (y))
= (((x)  (x))  ((y)  (y))
= ( )(x)  ( )(y).
( )(xy) ≥ ( )(x)  ( )(y).
ii. ( )(x-1
) = (x-1
)   (x-1
)
= (x)   (x)
= ( )(x).
( )(x-1
) = ( )(x)
iii. ( )(xy) = (xy)   (xy)
≥ ((x)  (y))  ((x)  (y))
= (((x)  (x))  ((y)  (y))
= ( )(x)  ( )(y).
( )(xy) ≥ ( )(x)  ( )(y).
iv. ( )(xy) = (xy)   (xy)
≥ ((x)  (y))  ((x)  (y))
= (((x)  (x))  ((y)  (y))
= ( )(x)  ( )(y).

( )(xy) ≥ ( )(x)  ( )(y).
Hence,  is a L-fuzzy sub  -group of G.
Remark
Arbitrary intersection of a L-fuzzy sub  -group of G is a L- fuzzy sub  -group of G.
3.5 Theorem
 is a L-fuzzy sub  -group of G iff c
is an anti L-fuzzy sub  -group of G.
Proof
Let  be a L-fuzzy sub  -group of G and for x , y  G, we have
i. (xy) ≥ (x)  (y)
 1 - c
(xy) ≥ ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ 1- ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ c
(x)  c
(y).
ii. (x-1
) = (x) for any x , y  G.
 1 - c
(x-1
) = 1- c
(x)
 c
(x-1
) = c
(x).
iii. (xy) ≥ (x)  (y)
 1-c
(xy) ≥ ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ 1- ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ c
(x)  c
(y).
iv. (xy) ≥ (x)  (y)
 1-c
(xy) ≥ ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ 1- ( 1-c
(x))  (1-c
(y))
 c
(xy) ≤ c
(x)  c
(y).
Hence, c
is an anti L-fuzzy sub  -group of G.
IV. PROPERTIES OF LEVEL SUBSETS OF A L-FUZZY SUB  -GROUP OF G
In this section, we introduce the concept of level subsets of a L-fuzzy sub  -group of G and discussed
some of its properties.
4.1 Definition
Let  be a L-fuzzy sub  -group of G. For any t [0,1], we define the set
U (  ; t) = { xG / (x) ≥ t } is called a upper level subset or a level subset of .
4.1 Theorem
Let  be a L-fuzzy sub  -group of G. Then for t  [0, 1] such that t ≤ (e), U (  ; t) is a sub  -
group of G.
Proof
For any x, y  U (  ; t) , we have,
(x) ≥ t ; (y) ≥ t.
Now, (xy−1
) ≥ (A)  (B)
(xy−1
) ≥ t  t
 (xy−1
) ≥ t.
xy−1
 U (  ; t).
Hence, U (  ; t) is a sub  -group of .
4.2 Theorem
Let  be a L-fuzzy subset of G such that U(  ; t) is a sub  -group of G. For t  [0,1] t ≤ (e),  is a
L-fuzzy sub  -group of G.
Proof
Let x, y  G and (x) = t1 and (y) = t2 .
Suppose t1 < t2 , then x , y  U(  ; t1).
As U (  ; t1) is a sub  -group of G, xy−1
 U(  ; t1).
Hence, (xy−1
) ≥ t1 = t1  t2

≥ (x)  (y)
That is, (xy−1
) ≥ (x)  (y).
By Theorem 3.2,  is a L-fuzzy sub  -group of G.
4.2 Definition
Let  be a L-fuzzy sub  -group of G. The sub  -groups U (  ; t ) for t [0,1] and t ≤ (e), are
called level sub  -groups of .
4.3 Theorem
Let  be a L-fuzzy sub  -group of G. If two level sub  -groups U (  ; t1), U(  ; t2), for, t1,t2  [0,1]
and t1 , t2 ≤ (e) with t1 < t2 of  are equal then there is no x  G such that t1 ≤ (x) < t2.
Proof
Let U(  ; t1) = U( ; t2).
Suppose there exists x  G such that t1 ≤ (x) < t2 , then U(  ; t2)  U ( ; t1).
Then x  U(  ; t1) , but x  U(  ; t2), which contradicts the assumption that, U(  ; t1) =
U(  ; t2). Hence there is no x  G such that t1 ≤ (x) < t2 .
Conversely, suppose that there is no x  G such that t1 ≤ (x) < t2 ,
Then, by definition, U(  ; t2)  U ( ; t1).
Let x  U(  ; t1) and there is no x  G such that t1 ≤ (x) < t2 .
Hence, x  U(  ; t2) and U(  ; t1)  U(  ; t2).
Hence, U(  ; t1) = U(  ; t2).
4.4 Theorem
A L-fuzzy subset  of G is a L-fuzzy sub  -group of G if and only if the level subsets U(  ; t), t  Image ,
are sub  -groups of .
Proof It is clear.
4.5 Theorem
Any sub  -group H of G can be realized as a level sub  -group of some L-fuzzy sub  -group of G.
Proof
Let  be a L-fuzzy subset and x  G.
Define,
t if x  H, where t ( 0,1].
 (x) =
0 if x  H ,
We shall prove that  is a L-fuzzy sub  -group of G.
Let x , y  G.
i. Suppose x, y  H, then xy  H , xy-1
 H , xy  H and xy  H.
(x) = t , (y) = t, (xy-1
) = t , ( xy ) = t and ( xy ) = t.
Hence (xy-1
) ≥ (x)  (y)
(xy) ≥ (x)  (y),
(xy) ≥ (x)  (y).
ii. Suppose x  H and y  H, then xy  H and xy-1
 H.
(x) = t, (y) = 0 and (xy-1
) = 0.
Hence (xy-1
) ≥ (x)  (y).
iii. Suppose x,y  H, then xy-1
 H or xy-1
 H.
(x) = 0, (y) = 0 and (xy-1
) = t or 0.
Hence (xy-1
) ≥ (x)  (y).
Thus in all cases,  is a L-fuzzy sub  -group of G.
For this L-fuzzy sub  -group, U(  ; t ) = H.

Remark
As a consequence of the Theorem 4.3 and Theorem 4.4, the level sub  -groups of a L-fuzzy sub  -
group  of G form a chain. Since (e) ≥ (x) for any x  G and therefore, U( ; t0 ) , where (e) = t0 is the
smallest and we have the chain :
{e}= U(  ; t0)  U(  ; t1 )  U( ; t2 )  …  U( ; tn ) = , where t0 > t 1 > t2 >…… >
tn.
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