Makgeolli Structures and Its Application in BCK/BCI-Algebras

Abstract

A fuzzy set is an extension of an existing set using fuzzy logic. Soft set theory is a generalization of fuzzy set theory. Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. The aim of this article is to introduce the concept of makgeolli structures using fuzzy and soft set theory and to apply it to BCK/BCI-algebras. The notion of makgeolli algebra and makgeolli ideal in BCK/BCI-algebras is defined, and several properties are investigated. It deals with the relationship between makgeolli algebra and makgeolli ideal, and several examples are given. Characterization of makgeolli algebra and makgeolli ideal are discussed, and a new makgeolli algebra from old one is established. A condition for makgeolli algebra to be makgeolli ideal in BCK-soft universe is considered, and we give example to show that makgeolli ideal is not makgeolli algebra in BCI-soft universe. Conditions for makgeolli ideal to be makgeolli algebra in BCI-soft universe are provided.

Positive Implicative Makgeolli Ideals of BCK-algebras

“…for all a, c ∈ E and x, z ∈ X by (I3) and (2). It follows from Theorem 3 that M (X,E) := (M E , G E , ξ) is a positive implicative makgeolli ideal of (X, E).…”

“…Fuzzy and soft set theory are good mathematical tools for dealing with uncertainty in a parametric manner. Ahn et al [2] introduced the concept of makgeolli structures as a hybrid structure using fuzzy and soft set theory, and applied it to BCK/BCI-algebras. In this article, we introduced the notion of a positive implicative makgeolli ideal in BCK-algebras, and investigated its properties.…”

“…(see [9-13, 15-17, 22, 23]). In 2019, Ahn et al [2] introduced the notion of makgeolli structures as a hybrid structure based on fuzzy set and soft set theory, and applied it to BCK/BCI-algebras.…”