Many-valued rough sets based on tied implications

Abstract

We investigate a general many-valued rough set theory, based on tied adjointness algebras, from both constructive and axiomatic approaches. The class of tied adjointness algebras constitutes a particularly rich generalization of residuated algebras and deals with implications (on two independently chosen posets (L,≤L) and (P,≤P), interpreting two, possibly different, types of uncertainty) tied by an integral commutative ordered monoid operation on P. We show that this model introduces a flexible extension of rough set theory and covers many fuzzy rough sets models studied in literature. We expound motivations behind the use of two lattices L and P in the definition of the approximation space, as a generalization of the usual one-lattice approach. This new setting increase the number of applications in which rough set theory can be applied.