On three types of $L$-fuzzy $β$-covering-based rough sets
This work addresses theoretical foundations in fuzzy rough set theory, which is incremental for researchers in mathematical logic and data analysis.
The paper tackles the problem of constructing and analyzing three types of L-fuzzy β-covering-based rough set models, resulting in the development of axiom sets, matrix representations, and interdependency conditions for these operators.
In this paper, we mainly construct three types of $L$-fuzzy $β$-covering-based rough set models and study the axiom sets, matrix representations and interdependency of these three pairs of $L$-fuzzy $β$-covering-based rough approximation operators. Firstly, we propose three pairs of $L$-fuzzy $β$-covering-based rough approximation operators by introducing the concepts such as $β$-degree of intersection and $β$-subsethood degree, which are generalizations of degree of intersection and subsethood degree, respectively. And then, the axiom set for each of these $L$-fuzzy $β$-covering-based rough approximation operator is investigated. Thirdly, we give the matrix representations of three types of $L$-fuzzy $β$-covering-based rough approximation operators, which make it valid to calculate the $L$-fuzzy $β$-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 $β$-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 $β$-coverings can generate the same lower and upper rough approximation operations.