Properties Of Derivations On KU-Algebra

Abstract

As it is well known, BCK and BCI-algebras are two classes of algebras of logic. They were introduced by Imai and Iseki [21, 22,26 ] and have been extensively investigated by many researchers. It is known that the class of BCK-algebras is a proper sub class of the BCI-algebras. The essential difference between BCK-algebras and BCIalgebras lies in the following: The Element 0 is the least element in BCK-algebras, while it is a minimal element in BCIalgebras .The class of all BCK-algebras is a quasivariety. Is´eki posed an interesting problem (solved by Wro´nski [ 60]) whether the class of BCKalgebras is a variety. In connection with this problem, Komori [39 ] introduced a notion of BCC-algebras, and Dudek [ 15] redefined the notion of BCC-algebras by using a dual form of the ordinary definition in the sense of Komori. Dudek and Zhang [16 ] introduced a new notion of ideals in BCC-algebras and described connections between such ideals and congruences . Prabpayak and Leerawat [55,56 ] introduced a new algebraic structure which is called KU-algebras. They introduced the concept of homomorphisms of KU-algebras and investigated some related properties. Zadeh [63 ] introduced the notion of fuzzy sets.the study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy groups by Rosenfeld [57 ].Xi [61] introduced the notion of fuzzy Bck-algebra. At present this concept has been applied to many mathematical branches, such as groups, functional analysis, probability theory and topology. Mostafa et al [46 ] introduced the notion of fuzzy KU-ideals of KU-algebras and then they investigated several basic properties which are related to fuzzy KU-ideals. Akram et al and Yaqoob et al [ 2, 62 ] introduced the notion of cubic sub-algebras and ideals in KU-algebras. They discussed relationship between a cubic subalgebra and a cubic KU-ideal. Several authors [7,8,10 ,11, 20 , 37 ] have studied derivations in rings and near rings. Jun and Xin [ 31 ] applied the notion of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular derivation in BCI - algebras. They investigated some of its properties, defined a d -derivation ideal and gave conditions for an ideal to be d-derivation