On three types of L-fuzzy β-covering-based rough sets

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.