On the structure of fuzzy variable precision rough sets based on generalized residuted lattices

This paper is devoted to propose generalized L-fuzzy variable precision rough sets, based on generalized residuated lattices with conjunctions not necessarily commutative over two universes, as a generalization of the notion of L-fuzzy variable precision rough sets based on residuated lattices with commutative conjunctions over one universe only. Then we define and investigate several classes of generalized L-fuzzy variable precision rough sets. The topological properties and granular representation of generalized L-fuzzy variable precision rough sets are also given. Our main concern is to propose a real number valued function for each approximation operator in generalized L-fuzzy variable precision rough sets and to measure its approximating ability. We observe that the functions of lower and upper approximation operators are natural generalizations of belief and plausibility functions respectively as given in the evidence theory. Using these functions, accuracy measure and degree of roughness is defined for generalized L-fuzzy variable precision rough sets. As an application of generalized L-fuzzy variable precision rough sets, I-fuzzy variable precision rough sets are proposed on the unit interval I = [0, 1], based on generalized residuated lattices induced by left-continuous pseudo-t-norms.