Upper rough approximation operators of quantale-valued similarities related to fuzzy orderings

Abstract

Let L be a commutative unital quantale. For every L-fuzzy relation E on a nonempty set X, we define an upper rough approximation operator on LX, which is a fuzzy extension of the classical Pawlak upper rough approximation operator. We show that this operator has close relation with the subsethood operator on X. Conversely, by an L-fuzzy closure operator on X, we can easily get an L-fuzzy relation. We show that this relation can be characterized by more smooth ways. Without the help of the lower approximation operator, L-fuzzy rough sets can still be studied by means of constructive and axiomatic approaches, and L-fuzzy similarities and L-fuzzy closure operators are one-to-one corresponding. We also show that, the L-topology induced by the upper rough approximation operator is stratified and Alexandrov.