A Study on Fuzzy Ideals of -Groups

Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of ( )-fuzzy ideals of -groups. These fuzzy ideals are characterized by their level ideals, and some other related properties are investigated. 1. Introduction and Basic Definitions The concept of a fuzzy set was introduced by Zadeh [1] in 1965, utilizing what Rosenfeld [2] defined as fuzzy subgroups. This was studied further in detail by different researchers in various algebraic systems. The concept of a fuzzy ideal of a ring was introduced by Liu [3]. The notion of fuzzy subnear-ring and fuzzy ideals was introduced by Abou-Zaid [4]. Then in many papers, fuzzy ideals of near-rings were discussed for example, see [5–11]. In [12], the idea of fuzzy point and its belongingness to and quasi coincidence with a fuzzy set were used to define -fuzzy subgroup, where , take one of the values from , . A fuzzy subgroup in the sense of Rosenfeld is in fact an -fuzzy subgroup. Thus, the concept of -fuzzy subgroup was introduced and discussed thoroughly in [7]. Bhakat and Das [13] introduced the concept of -fuzzy subrings and ideals of a ring. Davvaz [14, 15], Narayanan and Manikantan [16], and Zhan and Davvaz [17] studied a new sort of fuzzy subnear-ring (ideal and prime ideal) called -fuzzy subnear-ring (ideal and prime ideal) and gave characterizations in terms of the level ideals. In [18, 19], the idea of fuzzy ideals of -groups was defined, and various properties such as fundamental theorem of fuzzy ideals and fuzzy congruence were studied, respectively. In the present paper, we extend the idea of -fuzzy ideals of near-rings to the case of -groups and introduce the idea of fuzzy cosets with some results. We first recall some basic concepts for the sake of completeness. By a near-ring we mean a nonempty set with two binary operations “+” and “ ” satisfying the following axioms:(i) is a group,(ii) is a semigroup, (iii) for all .It is in fact a right near-ring because it satisfies the right distributive law. We will use the word “near-ring” to mean “right near-ring.” is said to be zero symmetric if for all . We denote by . Note that the missing left distributive law, , has to do with linearity if is considered as a function. Example 1. Let be a group, and let be the set of all mappings from into . We define + and on by Then, is a near-ring. Just in the same way as -modules or vector spaces are used in ring theory, -groups are used in near-ring theory. By an -group we mean a nonempty set