New Type of Soft (Prime) Ideals in Commutative BCK-Algebras

Abstract

The soft set theory is an important mathematical tool for dealing with uncertainty. By endowing a par-ameter set as a commutative BCK-algebra (that is commutative weak-BCI-algebra), the notions of a new type of soft prime ideals, annihilators of soft sets and new type of involutory soft ideals in commutative BCK-algebras are introduced. Two new compositional operations are defined and used to characterize the new type of soft ideals in commutative BCK-algebras. By using partial ordering on commutative BCK-algebras, some properties of the new type of soft ideals are studied. Properties of annihilators of soft sets and new type of involutory soft ideals are obtained. The existence of a new type of soft prime ideals in commutative BCK-algebras and its difference from the standard soft prime ideals are illustrated with examples. It is shown that a soft set is a new type of soft prime ideals in commutative BCK-algebras and its level set is a prime ideal is not a necessary and sufficient condition, which is different from the results of the usual fuzzy algebra. Some equivalent characterizations of the new type of soft prime ideals in commutative BCK-algebras are given. Furthermore, the properties of its homomorphism image and inverse image are discussed.

Keywords

• |bck-algebras|soft sets|prime ideals|annihilators|compositional operation