Contributions to Generalized Fuzzy Ideals of Semigroups

Abstract

The notion of ideals created by Dedekind for the theory of algebraic numbers was generalized by Emmy Neother for associative rings. The one- and two sided ideals introduced by her, are still central concepts in ring theory. Since then many papers on ideals for rings and semigroups appeared showing the importance of the concept. The fundamental concept of fuzzy set, introduced by Zadeh in his seminal paper [49] m 1965, provides a natural framework for generalizing several basic notions of algebra. Many papers on fuzzy sets appeared showing the importance of the concept and its application to logic, set theory, ring theory, group theory, semigroup theory, topology etc. Rosenfeld in [40] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroup. Kuroki initiated the theory of fuzzy semigroups (see [27-28]). A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [31], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines and fuzzy languages. Bhakat and Das (see [4-8]) gave the concept of (a, /3) - fuzzy subgroups of a group using the concept of 'belongs to' and 'quasi-coincident with' between a fuzzy point and a fuzzy set which is mentioned in [37]. They studied (E,E vq)- fuzzy subgroup of a group. In fact (E,E vq) -fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. Shabir et al. in [42] characterized regular semigroups by the properties of (E,E vq) -fuzzy ideals. Dudek et al. in [14] and Ma and Zhan in [30] defined (a, /3) -fuzzy ideals in hemirings, and investigated some related properties of hemirings. In [25], Jun and Song initiated the study of (a,/3) -fuzzy interior ideals of a semigroup. Kazanci and Yamak in [26], studied (E,E vq) -fuzzy bi-ideals of a semigroup. In [23] Jun et al. discussed (E,E vqk)-fuzzy h-ideals and (E,E vqk ) -fuzzy k - ideals of a hemiring. More detailed results on (E, E V q k) -type fuzzy ideals in hemirings can be seen in [1]. Shabir et al. in [43] characterized different classes of semigroups by (E,Evqk) -fuzzy ideals and (E,Evqk) -fuzzy bi-ideals. Ma et al. in [31], introduced the concept of (Ey,Ey vqo)-fuzzy ideals of Bel-algebras. Rehman and Shabir discussed (Ey,Ey vqo)-fuzzy ideals of ternary semigroups in [39].