Implicative Neutrosophic Quadruple BCK-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 B C K / B C I -numbers, and studied neutrosophic quadruple B C K / B C I -algebras. As a continuation of Jun et al.’s paper, the notion of implicative neutrosophic quadruple B C K -algebras is introduced, and several properties are investigated. Given a set Y, conditions for the neutrosophic quadruple Y-set N q ( Y ) to be a neutrosophic quadruple B C I -algebra are provided. Conditions for the neutrosophic quadruple Y-set N q ( Y ) to be an implicative neutrosophic quadruple B C K -algebra are provided. Given subsets I and J of a B C K -algebra Y, conditions for the neutrosophic quadruple ( I , J ) -set N q ( I , J ) to be an implicative ideal of the neutrosophic quadruple B C K -algebra N q ( Y ) are discussed.

Generalized Ideals of BCK/BCI-Algebras Based on Fuzzy Soft Set Theory

“…Larimi and Jun introduced the concepts of (∈, ∈∨q)-intuitionistic fuzzy h-ideals of hemiring [22]. Various attributes of BCK/BCI-algebra are considered in [23][24][25][26][27][28][29][30][31].…”

Fuzzy soft set theory applied to commutative ideals of BCK-algebras

“…The ideal theory in algebraic structures has been studied in , Muhiuddin et al (2020d;, Senapati et al (2019), and Talee et al (2020). Some important generalizations of fuzzy sets were applied in / -algebras (Muhiuddin and Aldhafeeri, 2019;Muhiuddin and Al-Roqi, 2016;2018;Muhiuddin and Jun, 2019;Muhiuddin et al, 2017b;2017c;2019b;2019c;2019d;2020c;Muhiuddin, 2019).…”

Primeness of Relative Annihilators in BCK-Algebra

“…They provide the conditions that the relative annihilator of an ideal with respect to an ideal needs to be ideal, and discussed conditions for the relative annihilator ideal to be an implicative (resp., positive implicative, commutative) ideal. Moreover, in some articles, different properties of ideals in logical algebras and ordered algebraic structures were concerned (see [9][10][11][12][13][14][15][16][17][18]). In order to investigate these kinds of properties for an arbitrary ideal in BCI/BCK-algebra, we need to know about the decomposition of an ideal.…”