FUZZY IDEALS OF SOME ALGEBRAS

Abstract

The notion of BCK-algebras introduced the class of abstract algebras BCK-algebras as a generalization of the concept of set-theoretic difference and propositional calculus . In 1966, (Y. Iami and K. Iseki) introduced the notion of BCK-algebra, and the notion of BCI-algebra which is a generalization of BCK-algebra ([23], [25]). Since the numerous mathematical papers have been written investigating the algebraic properties of the BCK I BCI-algebras , there is a great deal of literature has been produced on the theory of BCKIBCI-rugebras, emphasis seems to have been put on the ideal theory of BCK/ BCI-algebras ([23], [25],[31]). In 1983, (Q. Hu and X. Li) ([21],[22]) introduced the notion of a BCH-algebra which is a generalization of the notion ofBCI-algebras and studied a few properties of these algebras. In 1983 , (Y. Komori) [46] introduced a notion of BCC-algebras, and (W. A. Dudek) ([16],[17]) redefined the notion ofBCC-algebras by using a dual form of the ordinary definition in the sense of Y. Komori. In [13] ,(W. A. Dudek and X. H. Zhang) introduced a notion of BCC-ideals in BCC algebras and described connections between such ideals and congruences. The concept of fuzzy sets was fust introduced by (L.A. Zadeh) [78],in 1965 . From that time, the theory of fuzzy sets has been developed in many directions and found applications in a wide variety of fields . It application to various mathematical contexts has given rise to what is now commonly called fuzzy mathematics. Fuzzy algebras is an important branch of fuzzy mathematics, they have studied some algebraic structures such as fuzzy groups by (A. Rosenfeld) [69] in 1971 . In the time , , they have studied some algebraic structures such as fuzzy rings, fuzzy ideals, fuzzy vectors spaces. In 1975 , (L.A. Zadeh) [79] introduced the notion of an interval-valued fuzzy BCK-ideal (briefly i-v fuzzy BCK-ideal). In 1991, (0. Xi) [74] applied the concept of fuzzy BCK-algebras . (J. Meng and Y.B. Jun) [52] in 1994, studied the fuzzification ofBCK-algebra and BCI-algebra. In 2001 (Y. B. Jun) [13] introduced fuzzy BCC-subalgebras and investigated some related properties.