Fuzzy Structures of Some Algebras

Abstract

The notion of BCK-algebras was proposed by Iami and Iseki [16,17]. In 1966 Iseki [13,15] introduced the notion of a BCI-algebra ,which is a generalization of BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK / BCI-algebras and their relationship with other structures including lattices and Boolean algebras. There is a great deal of literature which has been produced on the theory of BCK/BCIalgebras, in particular, emphasis seems to have been put on the ideal theory of BCK/BCI-algebras [13,26,42,44,45,52,58]. For the general development of BCK/BCI-algebras the ideal theory plays an important role. The concept of fuzzy sets was first introduced by Zadeh [53]. From that time, the theory of fuzzy sets which has been developed in many directions and found applications in a wide variety of fields [1,2,7, 8 ,9 ,10,11 ,13,18, 22,23,40,41,49,56]. In 1991, Xi [49] applied the concept of fuzzy sets to BCI, BCK, MV-algebras. The ideal theory and its fuzzification play an important role. In [42] J. Meng and Y.B. Jun studied medial BCI-algebras. In [47] S.M. Mostafa, Y.B. Jun and A. El-Menshawy introduce the notion of medial ideals in BCI-algebras, they state the fuzzification of medial ideals and investigate its properties. There are several kinds of fuzzy sets extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, interval valued fuzzy sets, vague sets etc. Biswas in [9] gave the idea of anti fuzzy subgroups. Zadeh [53] made an extension of the concept of fuzzy set by an interval-valued fuzzy set (i.e., a fuzzy set with an interval valued membership function). Summary - 2 - In [10], Biswas defined interval valued fuzzy subgroups and investigated some elementary properties. Jun [25] defined a doubt fuzzy sub-algebra, doubt fuzzy ideal, in BCI/ BCK -algebras and got some results about it. The idea of “intuitionistic fuzzy set” was first published by Atanassov [4,5] as a generalization of the notion of fuzzy sets. After that many researchers consider the Fuzzifications of ideals and sub-algebras in BCK/BCI-algebras. [18,46] introduced the notion of intuitionistic fuzzy (medial)- ideals and investigated some simple but elegant results. [28,39] discussed fuzzy translations, (normalized, maximal) fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BCK/BCI-algebras and introduced the relations among fuzzy translations, (normalized, maximal) fuzzy extensions and fuzzy multiplications.