Abstract: Concept of topological group and some of

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.

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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

x

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