On three types of $L$-fuzzy $\beta$-covering-based rough sets

TL;DR

This paper develops three new $L$-fuzzy $eta$-covering-based rough set models, exploring their axioms, matrix representations, and interdependencies, advancing the theoretical framework of fuzzy rough set theory.

Contribution

It introduces three types of $L$-fuzzy $eta$-covering-based rough approximation operators with new concepts, axioms, and matrix methods, and analyzes their interdependencies.

Findings

Proposed three pairs of $L$-fuzzy $eta$-covering-based rough approximation operators.

Established axioms and matrix representations for these operators.

Determined conditions for when different coverings produce equivalent rough approximations.

Abstract

In this paper, we mainly construct three types of $L$-fuzzy $\beta$-covering-based rough set models and study the axiom sets, matrix representations and interdependency of these three pairs of $L$-fuzzy $\beta$-covering-based rough approximation operators. Firstly, we propose three pairs of $L$-fuzzy $\beta$-covering-based rough approximation operators by introducing the concepts such as $\beta$-degree of intersection and $\beta$-subsethood degree, which are generalizations of degree of intersection and subsethood degree, respectively. And then, the axiom set for each of these $L$-fuzzy $\beta$-covering-based rough approximation operator is investigated. Thirdly, we give the matrix representations of three types of $L$-fuzzy $\beta$-covering-based rough approximation operators, which make it valid to calculate the $L$-fuzzy $\beta$-covering-based lower and upper rough approximation operators through operations on matrices. Finally, the interdependency of the three pairs of rough approximation operators based on $L$-fuzzy $\beta$-covering is studied by using the notion of reducible elements and independent elements. In other words, we present the necessary and sufficient conditions under which two $L$-fuzzy $\beta$-coverings can generate the same lower and upper rough approximation operations.