  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.
