Fuzzy rough sets based on generalized residuated lattices

This paper is devoted to propose generalized L-fuzzy rough sets as a further generalization of the notion of L-fuzzy rough sets. A quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non-commutative. Generalized L-fuzzy rough sets are characterized from both constructive and axiomatic approaches. In the constructive approach, various classes of generalized L-fuzzy rough sets are investigated. Moreover, the relationship between generalized L-fuzzy rough sets and L-topologies on an arbitrary universe is discussed with generalized lower and upper sets. As an application of generalized L-fuzzy rough sets, fuzzy rough sets are proposed and studied on the unit interval [0, 1], which are based on generalized residuated lattices induced by left-continuous pseudo-t-norms.