Sun Shin Ahn, Young Joo Seo, Young Bae Jun

Generalized Lukasiewicz fuzzy subalgebras of BCI-algebras and BCK-algebras

Introduction

Generalized lukasiewicz fuzzy subalgebras of bci-algebras and bck-algebras. Generalize Lukasiewicz fuzzy subalgebras in BCI/BCK-algebras. Define and explore (α,ϵ)-Lukasiewicz fuzzy subalgebras, their properties, relationships, and characterizations using fuzzy points.

Abstract

The aim of this paper is to generalize Lukasiewicz fuzzy subalgebras in BCK/BCI-algebras. First, the concept of (α,ϵ)-Lukasiewicz fuzzy subalgebras using fuzzy points is defined and examples to explain it are given, and then several properties are investigated. The relationship between Lukasiewicz fuzzy subalgebras and (α,ϵ)-Lukasiewicz fuzzy subalgebras is discussed, and the conditions under which the ϵ-Lukasiewicz fuzzy set to be an (α,ϵ)-Lukasiewicz fuzzy subalgebra are explored. The characterizations of (α,ϵ)-Lukasiewicz fuzzy subalgebras are examined. Conditions under which Lukasiewicz ∈-set, Lukasiewicz q-set and Lukasiewicz O-set can be subalgebras are handled.

Review

This paper presents a significant theoretical advancement in the field of fuzzy algebraic structures by introducing and thoroughly investigating the concept of (α,ϵ)-Lukasiewicz fuzzy subalgebras within BCK/BCI-algebras. The primary objective of generalizing existing Lukasiewicz fuzzy subalgebras is effectively achieved through a rigorous definition utilizing fuzzy points, which is then substantiated by illustrative examples to enhance clarity. This systematic approach in extending fundamental concepts of fuzzy set theory to non-classical logic algebras represents a valuable contribution to the ongoing development of fuzzy mathematics. The authors meticulously explore various facets of these newly defined structures. A core component of the research involves a detailed discussion on the relationship between the original Lukasiewicz fuzzy subalgebras and their generalized (α,ϵ) counterparts. Furthermore, the paper provides a comprehensive analysis of the specific conditions under which an ϵ-Lukasiewicz fuzzy set can qualify as an (α,ϵ)-Lukasiewicz fuzzy subalgebra, along with an in-depth characterization of these structures. The study also extends to examining the conditions for Lukasiewicz ∈-set, Lukasiewicz q-set, and Lukasiewicz O-set to form subalgebras, demonstrating a robust and exhaustive exploration of the theoretical implications. The theoretical contributions of this paper are substantial for researchers working in fuzzy algebra, logic, and related computational fields. By offering a more generalized framework, the work potentially paves the way for greater flexibility in modeling uncertainty within BCK/BCI-algebras and could inspire further research into other generalized fuzzy algebraic structures. The clear definitions, systematic property investigations, and thorough characterizations make this a well-structured and impactful paper that solidifies the foundational understanding of generalized fuzzy subalgebras, offering a strong basis for future theoretical developments and potential applications.

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