Implicative Neutrosophic Quadruple <i>BCK</i>-Algebras and Ideals

Abstract

A neutrosophic set is initiated by Smarandache, and it is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache. Using this idea, Jun et al. introduced the notion of neutrosophic quadruple BCK/BCI-numbers, and studied neutrosophic quadruple BCK/BCI-algebras. As a continuation of Jun et al.' s paper, the notion of implicative neutrosophic quadruple BCK-algebras is introduced, and several properties are investigated. Given a set Upsilon, conditions for the neutrosophic quadruple Upsilon-set N-q (Upsilon) to be a neutrosophic quadruple BCI-algebra are provided. Conditions for the neutrosophic quadruple Upsilon-set N-q(Upsilon) to be an implicative neutrosophic quadruple BCK-algebra are provided. Given subsets I and J of a BCK-algebra Upsilon, conditions for the neutrosophic quadruple (I, J)-set N-q (I, J) to be an implicative ideal of the neutrosophic quadruple BCK-algebra N-q (Upsilon) are discussed.