-fuzzy Sub- Commutative Ideals in Bci-algebras

The concept of fuzzy set, which was published by Zadeh in his classic paper [24] of 1965, was applied by many researchers to generalize some of the basic concepts of algebra. The fuzzy algebraic structures play a central role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, real analysis, measure theory etc. In 1991, Xi applied fuzzy subsets in BCK-algebras [23] and got some interesting results. In 1993, Ahmad [1] and Jun [12] applied the concept of fuzzy sets to BCIalgebras. The concept of sub-commutative ideals in a BCI-algebra was initiated by Liu and Meng [14]. Jun studied fuzzy sub-implicative ideals of BCI-algebras in [9]. Liu et al. [15] discussed FSI-ideals and FSC-ideals of BCI-algebras. In [8], Hedayati studied connections between generalized fuzzy ideals and subimplicative ideals of BCI-algebras. Zhan et al. discussed characterizations of generalized fuzzy ideals of BCI-algebras in [26]. In 1971, Rosenfeld [22] laid the foundations of fuzzy groups. In [20], Murali defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. The idea of quasi-coincidence of a fuzzy point with a fuzzy set given in [21], plays a vital role to generate some different types of fuzzy subgroups, called (α, β)-fuzzy subgroups, introduced by Bhakat and Das [4]. In particular, ) , ( q ∨ ∈ ∈ -fuzzy subgroup is an important and useful generalization of the Rosenfeld’s fuzzy subgroup. Bhakat [2, 3] studied