Characterizations of Abel-Grassmann's Groupoids by Intuitionistic Fuzzy Ideals

An AG-groupoid is a non-associative algebraic structure mid way between a groupoid and a commutative semigroup. The left identity in an AGgroupoid if exists is unique [9]. An AG-groupoid is non-associative and non-commutative algebraic structure, nevertheless, it posses many interesting properties which we usually find in associative and commutative algebraic structures. An AG-groupoid with right identity becomes a commutative monoid [9]. An AG-groupoid is basically the generalization of semigroup [5] with wide range of applications in theory of flocks [10]. The theory of flocks tries to describe the human behavior and interaction. The concept of fuzzy sets was first proposed by Zadeh [12] in 1965. Several researchers were conducted on the generalization of the notion of fuzzy set. Given a set S, a fuzzy subset of S is, by definition an arbitrary mapping f : S [0,1], → where [0,1] is the unit interval. A fuzzy subset is a class of objects with grades of membership. A. Rosenfeld [11] was the first who consider the case of a groupoid in terms of fuzzy sets. Kuroki has been first studied the fuzzy sets on semigroups [7]. As an important generalization of the notion of fuzzy set, Atanassov [at], introduced the concept of an intuitionistic fuzzy set. De et al. [2] studied the Sanchez's approach for medical diagnosis and extended this concept with the notion of intuitionistic fuzzy set theory. Dengfeng and Chunfian [3] introduced the concept of the degree of similarity between intuitionistic fuzzy sets, which may be finite or continuous and gave corresponding proofs of this similarity measure and discussed applications of the similarity measures between intuitionistic fuzzy sets to pattern recognition problems. The concept of intuitionistic fuzzy AG-groupoid was first given by Khan and Faisal in [6]. For more details and applications one can see [13-17].