Couple fuzzy covering rough set models and their generalizations to CCD lattices

Abstract

We make gradual generalizations in this paper, from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy covering rough set models in fuzzy rough set theory, and further to the concepts of couple L-fuzzy covering rough set models in L-fuzzy rough set theory.Given a fuzzy covering approximation space (U,C˜) and β∈(0,1], for each x∈U, we divide C˜ into two parts ⊤x={C˜i∈C˜:C˜i(x)≥β} and ⊥x={C˜i∈C˜:C˜i(x)<β}. To fully describe x from positive aspect and negative aspect, both the two parts ⊤x and ⊥x are important, especially the combination of them. So, in this paper, based on the two parts, we define 1st couple β-fuzzy covering rough set models [(P−˜,P+˜),(Q−˜,Q+˜)] and 2nd couple fuzzy β-covering rough set models [(P−‾,P+‾),(Q−‾,Q+‾)]. Both the two types of couple fuzzy β-covering rough set models are generalizations of the twin approximations defined in covering rough set theory. Since each pair of operators in these models are not only closely related but complementary, they can be used to analyze and solve practical problems from positive and negative aspects so as to make a crucial decision. So we then give some examples to show their practical value. The relationships between our models and some other models introduced in previous literature are investigated, and the matrix methods are given to calculate the related approximations and to describe the relationships between every couple models.To generalize the couple fuzzy β-covering rough set models to the CCD lattice are of a bit complicated, because any two elements in the lattice cannot be compared with each other generally. After some effort we successfully construct the ideal couple L-fuzzy β-covering rough set models in L-fuzzy rough set theory which are just the generalizations of the two types of couple fuzzy β-covering rough set models. We also obtain the lattice matrix representations to calculate the related approximations.