INTUITIONISTIC FUZZY INTERIOR IDEAL OF SEMIGROUP BASED ON INTUITIONISTIC FUZZY POINT

The intuitionistic fuzzification of the notion of an interior ideal in ordered semigroups is considered. The purpose of this study is to introduce the notion of ) , ( q    -intuitionistic fuzzy interior ideals and ) , (   -intuitionistic fuzzy interior ideals of semigroups. The important milestone of the present paper is to link ordinary intuitionistic fuzzy interior ideals, ) , (   -intuitionistic fuzzy interior ideals and ) , ( q    -intuitionistic fuzzy interior ideals. Moreover, semigroups are characterized by the properties of these new notions.


ABSTRACT
The intuitionistic fuzzification of the notion of an interior ideal in ordered semigroups is considered.The purpose of this study is to introduce the notion of

INTRODUCTION
Fuzzy set theory introduced by Zadeh is a useful tool to describe situations in which the data are imprecise or vague and handle such situations by attributing a degree to which a certain object belongs to a set [1].Just after the establishment of fuzzy set theory, it has developed in several directions and is discovering applications in a wide variety of fields.Rosenfeld used this idea to develop the theory of fuzzy groups [2].Fuzzy set theory provides a natural framework for generalization of the basic notions of algebra.But there is no means to incorporate the hesitation or uncertainty in the membership degrees.Atanassov introduced the concept of intuitionistic fuzzy sets in 1986, which constitute an extension of a fuzzy set theory [3].Since then, many researchers have investigated this topic such as intuitionistic fuzzy group [4][5][6].Intuitionistic fuzzy sets give both a membership degree and a non-membership degree, where sum of membership degree and non-membership degree needs to be less or equal to 1 .Pu and Liu introduced the notions of "belongings" and "quasi- coincidence" of fuzzy point and fuzzy set [7].Bhakat and Das introduced ) , ( q    -fuzzy subgroup as a generalization of Rosenfeld's fuzzy subgroups by using the combine notions of "belongings ( )" and "quasi-coincidence ( q )" of fuzzy point and fuzzy set [8].In semigroup Jun and Song gave the concept of an ) , (   -fuzzy interior ideal, which is a generalization of a fuzzy interior ideal by using the idea of quasi-coincidence of a fuzzy point with a fuzzy set [9].In Davvaz and Khan discussed some characterizations of regular ordered semigroups in terms of ) , (   -fuzzy generalized bi-ideals, } and q    [10].Some authors have investigated similar type of generalizations of other algebraic structures.Kazanci and Yamak introduced the concept of a generalized fuzzy bi-ideal in semigroups and gave some properties of fuzzy bi-ideals in terms of -fuzzy bi-ideals [11].A group of scientists characterized regular semigroups by the properties of -fuzzy ideals, bi-ideals and quasi-ideals [12].In a research, author has introduced more generalized forms of (  , )-fuzzy ideals and defined -fuzzy ideals of semigroups [13].Kazanci and Yamak [11] gave -fuzzy interior ideals of ordered semigroups and investigated some characterizations of ordered semigroups in terms of -fuzzy interior ideals [14].The notion of -fuzzy ideals, generalized bi-ideals and quasi-ideals of a semigroup and characterized regular semigroups by the properties of these ideals [16].
Kim and Jun introduced the notion of an intuitionistic fuzzy interior ideal of a semigroup S and gave its characterization [5].In a study conducted by some researcher redefined ) , (   -intuitionistic fuzzy subgroups [6].
In this paper, we introduce the notion of -intuitionistic fuzzy interior ideals of a semigroup and discuss some interesting results and investigate the relationships among ordinary intuitionistic fuzzy interior ideals, ) , (   -intuitionistic fuzzy interior ideals and ) , ( q    intuitionistic fuzzy interior ideals of semigroups.

BASIC DEFINITIONS AND PRELIMINARIES
In what follows S will represent a semigroup unless otherwise stated A non-empty subset A of a semigroup S is called an interior ideal of An intuitionistic fuzzy subset (briefly IFS) A in a non-empty set X is an object having the form ) and the degree of non-membership (namely to the set A , respectively and for all X x  .For the sack of simplicity, we shall use the symbol be an IFS of S , then A is called an intuitionistic fuzzy subsemigroup of S if for all

Definition
, be an intuitionistic fuzzy subset of a semigroup S .Then A is called an intuitionistic fuzzy interior ideal of S if for all S a y x  , , the following conditions hold; of a semigroup S respectively are denoted and defined as is an interior ideal of S .

Definition
Let x be a point of a non-empty set is called an intuitionistic fuzzy point (IFP for short) in X , where t (resp.s ) is the degree of membership (resp.non- do not hold.

Definition
For a non-empty subset A of S the characteristic function

Proposition
A non-empty subset A of S is an intuitionistic fuzzy interior ideal of S if and only if the characteristic function of A is an intuitionistic fuzzy interior ideal of S .

-INTUITIONISTIC FUZZY INTERIOR IDEALS
In this section, we introduce -intuitionistic fuzzy interior ideals of a semigroup S .

Definition
An intuitionistic fuzzy subset , satisfies the following conditions:

Example
Consider a semigroup

Define an IFS  
 s s s .

Lemma
An intuitionistic fuzzy subset , a contradiction by ( ( , then . We consider the following two cases: Case I: If . Thus, in both case we see that .We consider the following two cases: . Thus, in both case we see that .

Definition
An intuitionistic fuzzy subset

Define an IFS  
 s s s .

Theorem
be an intuitionistic fuzzy subset of S .Then A is an intuitionistic fuzzy interior ideal of S if A is an fuzzy interior ideal of S .

Proof
Let A is an intuitionistic fuzzy interior ideal of S and From the Theorem (3.6) it is clear that every intuitionistic fuzzy interior ideal is an -intuitionistic fuzzy interior ideal and it is obvious that every ) , (   -intuitionistic fuzzy interior ideal of S is an intuitionistic fuzzy interior ideal of S .
In the next theorem, we give a condition for an fuzzy interior ideal of S to be an ) , (   -intuitionistic fuzzy interior ideal of S .

Theorem
intuitionistic fuzzy interior ideal of S .

Theorem
For an intuitionistic fuzzy subset From ( interior ideals of semigroups.The important milestone of the present paper is to link ordinary intuitionistic fuzzy interior ideals, interior ideals.Moreover, semigroups are characterized by the properties of these new notions. . A non-empty subset A of S is called a subsemigroup of S if Cite the article: Hidayat Ullah Khan, Nor Haniza Sarmin, Asghar Khan, and Faiz Muhammad Khan and Taj Muhammad Khan (2017).In tuitionistic Fuzzy Interior Ideal Of Semigroup Based On Intuitionistic Fuzzy Point.Matrix Science Mathematic, 1(2) : 22-26.

table : Table 1 :
Multiplication of fuzzy interior ideal of S it and only if for all fuzzy interior ideal of S , if for all is an interior ideal of S .Assume that there exist