Abstract:

Concept of topological group and some of their properties are

introduced in this manuscript. Fuzzy soft topological groups is established in

new way by combining topological groups and fuzzy soft set, finally some

definition and results are also studied.

Keyword:

Fuzzy sets, soft sets, fuzzy soft set, topological group, fuzzy

soft topological group.

Introduction:

In 1965, fuzzy sets and fuzzy set operations were introduced by Zadeh.

By using fuzzy set theory, some of people have been made to develop several

mathematical structures. In engineering, economics, medical science, social sciences and many

others there arise many problems which contain many uncertainties. The

classical methods become useless to solve these types of problems. To overcome

these uncertainties, in 1999, Molodtsov introduced a new mathematical tool

named as soft set theory for modeling

vagueness and uncertainties. Later on, many mathematician explored this theory. Combination of soft set theory and fuzzy set theory

to develop a new termed as “Fuzzy soft set”. In 2001-2003, some aspects of soft

set and fuzzy soft set was worked by Maji et al. It is natural to investigate

of topological structures in group. In 2014, fuzzy soft topological groups were

introduced by S.Nazmul, S.k Samanta, with combination of fuzzy soft topological

spaces and fuzzy soft group.

Preliminary

2.1 Fuzzy sets:

A fuzzy set is a pair (U,m) where U is a set and m:U ?0,1 a

membership function.

·

Let

x belong to U, then x is called not included in fuzzy set (U,m) if m(x)=1

·

Fully

included in fuzzy set (U,m) if m(x)=1

·

Partial

included if 0< m(x) <1
2.2 Soft sets:
Let U be a
universal set and E a set of parameters or attributes with respect to U. A is
subset of E. A pair (F,A) is called soft set over U, where F:A?P(U), P(U)
denotes power set of U.
Soft
Subset:
Let (F,A)
and (F,B) be two soft sets over U, then (F,A) soft subset of (F,B) if
(1)- A is subset of B
(2)- F(e)
and G (e) are identical approximation
Soft
Equal Set:
(F,A) is
soft equal set to (G,B) denoted by (F,A) =(F,B)
If
(a)
(F,A)
is soft subset of (G,B)
(b)
(G,B)
is soft subset of (F,A)
3. Fuzzy Soft Set:
A pair (F;A) is called a fuzzy soft set over U, where F is a mapping
given by F : A ?FP(U), that is, for each a 2 A, F(a) = Fa : X ? 0;1 is a fuzzy set
on U. A fuzzy soft set can be represented as
F(A) = {f(x, F(x)) : x
A; F(x)
FP(U)}
Fuzzy
soft subset:
Let (F1,A)
and (F2,A) be two fuzzy soft sets over a common universe X.Then (F1,A) is said
to be fuzzy soft subset of (F2,A) if F1(a)<_F2(a),v a €A. this relation is
denoted by (F1,A)
(F2,A)
Fuzzy
Soft Equal Set :
Let (F1,A) and
(F2,A) be two fuzzy soft sets over a common universe X.Then (F1,A) is said to be
fuzzy soft equal to (F2,A)
(a)
(F1,A)
is fuzzy soft subset of (F2,A)
(b)
(F2,A)
is fuzzy soft subset of (F1,A)
4-
Topological Groups:
A group is
said to be topological group, define a mapping
F: G
G
G
g: G
G
where
f(u,h)= uh
g(u)=
u,h G
denoted by
((G,.),t)
Example:
Discrete topology or the indiscrete topology
of any group is a topology group .
Example:
R is a
topological group uder addition.
Example:
C* and R*
under multiplication are topological group.
Preposition:
Every open
sub group of G is closed if G is a topological group.
5- Fuzzy
Soft Topological Group:
Let (X, A)
be a fuzzy soft set and G be a group. T be a fuzzy soft topology.
(X, A,
) is
said to be fuzzy soft topology group over G if the mapping
(1)- f:
(X,A,
) X
(X,A,
)
(G,A,
)
F(x,y) =xy
(2)- g :
(X,A,
)
(G,A,
)
g (x)=
where x,y€G