Operations, similarity measures, and rough-set approximations for uncertain sets

Tóm tắt

An uncertain set assigns a generalised uncertainty value to each element, thus providing a single membership-function template that encompasses many degree-based models, including fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic sets [1]. Building on this unified viewpoint, we investigate three core components of uncertainty-aware reasoning: algebraic operations, similarity measures, and rough-set-style approximations. First, for a fixed uncertainty model M whose degree domain is equipped with a De Morgan lattice structure, inclusion, complement, union, intersection, difference, and level-set (α-cut) constructions are defined for M-type uncertain sets. Second, the author introduces an axiomatic notion of an uncertain similarity measure and provides concrete families, including normalised L1- and Chebyshev-type similarities, together with basic properties and reductions to familiar formulas in classical special cases. Third, uncertain rough sets are defined via M-valued relations and corresponding lower/upper approximation operators, and the resulting framework is shown to subsume standard fuzzy rough sets and component-wise neutrosophic rough sets. Finally, a categorical extension based on functorial sets is developed, obtaining functorial rough sets whose approximation operators satisfy naturality with respect to mo