Derivations on some algebras

Abstract

In 1966, Y. Imai and K. Isèki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras [27, 28, 29, 30, 31, 32]. It is known that the class of BCK-algebras is proper subclass of the class of BCI-algebras. J. Neggers, et al. [55] introduced a notion, called Qalgebras, which is a generalization of BCH / BCI / BCK-algebras and generalized some theorems discussed in BCI-algebras. Moreover, Ahn and Kim [3] introduced the notion of QS-algebras which is a proper subclass of Q-algebras. Kondo [39] proved that, each theorem of QS-algebras is provable in the theory of abelian groups and conversely each theorem of abelian groups is provable in the theory of QS-algebras. In [17], W. A. Dudek and X. H. Zhang introduced a new notion of ideals in BCC-algebras and described connections between such ideals and congruences. In the theory of rings, the properties of derivations are important. The notion of the ring with derivation is quite old and plays a significant role in the integration of analysis, algebraic geometry and algebra. Several authors [5, 6, 9, 10, 36, 37, 41] have studied derivations in rings and near rings. Jun and Xin [35] 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. Later, Abujabal and Al-Shehri [2], defined a left derivation in BCI-algebras and investigated a regular left derivation. Zhan and Liu [65] studied fderivations in BCI-algebras and proved some results. Muhiuddin and Al-roqi [53] introduced the notion of ) , (  -derivation in a BCI-algebra and investigated related properties. They provided a condition for a ( , ) - derivation to be regular. They also introduced the concepts of a ( , ) d - invariant ( , ) -derivation and α- ideal, and then they investigated their relations. Furthermore, they obtained some results on regular ( , ) -derivations. Moreover, they studied the notion of t-derivations on BCIalgebras [54] and obtain some of its related properties. Further, they characterize the notion of p-semisimple BCI-algebra X by using the notion of t-derivation. C.Prabpayak ,U.Leerawat [57] applied the notion of a regular derivation to BCC-algebras and investigated some related properties. II The notion of fuzzy sets, has introduced by Zadeh [64] , whose applied to many mathematical branches. At present these ideas have been applied to other algebraic structures such as semi groups, groups, rings, ideals, modules, vector spaces, etc. In 1991 Xi [61] , applied the concept of fuzzy sets to BCK-algebras. In 1993, Ahmad and Jun applied the concept of fuzzy sets to BCI-algebras .For the general development of BCI(BCK)-algebras the fuzzy ideal theory plays an important role. In the ideal theory of BCI(BCK)-algebras, generation of an ideal by a subset in a BCI(BCK)-algebra is an important problem. W. A. Dudek and Y. B. Jun (cf. [16]) considered the fuzzification of BCC-ideals in BCC-algebras. They proved that every fuzzy BCC-ideal of a BCC-algebras is a fuzzy BCK-ideal, and showed that the converse is not true by providing a counter example. They also proved that in a BCC-algebras every fuzzy BCK-ideal is a fuzzy BCC-subalgebra, and that in a BCK-algebras the notion of a fuzzy BCK-ideal and a fuzzy BCC-ideal coincide. W. Dudek and Y. B. Jun [16] described several properties of fuzzy BCC-ideals in BCC-algebras, and discussed an extension of fuzzy BCC-ideals. Motivated by a lot of work done on derivations of BCI-algebras and derivations of other related algebraic structures and the fuzzy of algebraic structures, the thesis deals with the study of derivation of a QS-algebras and fuzzy left (right) derivations BCC-ideals in BCC-algebras. This thesis has been mainly divided into three chapters Chapter