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

Abstract

An uncertain set assigns a generalised uncertainty value to each element, thus providing a single membership-function template that encompasses many degreebased 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 roughset-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 (a-cut) constructions are defined for M-type uncertain sets. Second, the author introduces an axiomatic notion of an uncertain similarity measure and provide 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 componentwise 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 morphisms, thereby enabling the compositional transport of rough approximations across changing universes.