A novel axiomatic approach to L-valued rough sets within an L-universe via inner product and outer product of L-subsets

Abstract

The fuzzy rough approximation operator serves as the cornerstone of fuzzy rough set theory and its practical applications. Axiomatization is a crucial approach in the exploration of fuzzy rough sets, aiming to offer a clear and direct characterization of fuzzy rough approximation operators. Among the fundamental tools employed in this process, the inner product and outer product of fuzzy sets stand out as essential components in the axiomatization of fuzzy rough sets. In this paper, we will develop the axiomatization of a comprehensive fuzzy rough set theory, that is, the so-called L-valued rough sets with an L-set serving as the foundational universe (referred to as the L-universe) for defining L-valued rough approximation operators, where L typically denotes a GL-quantale. Firstly, we give the notions of inner product and outer product of two L-subsets within an L-universe and examine their basic properties. It is shown that these notions are extensions of the corresponding notion of fuzzy sets within a classical universe. Secondly, leveraging the inner product and outer product of L-subsets, we respectively characterize L-valued upper and lower rough approximation operators generated by general, reflexive, transitive, symmetric, Euclidean, and median L-value relations on L-universe as well as their compositions. Finally, utilizing the provided axiomatic characterizations, we present the precise examples for the least and largest equivalent L-valued upper and lower rough approximation operators. Notably, many existing axiom characterizations of fuzzy rough sets within classical universe can be viewed as direct consequences of our findings.