🤖 AI Summary
This study investigates the conditions under which the rough set algebra induced by a reflexive relation possesses the characteristic lattice and algebraic properties typically associated with equivalence relations. Employing tools from lattice theory, Dedekind–MacNeille completion, and pseudocomplemented Kleene algebras, the paper provides the first systematic characterization of necessary and sufficient conditions for the Dedekind–MacNeille completion of the rough set algebra, denoted DM(RS), to be a spatial and completely distributive lattice, a regular pseudocomplemented Kleene algebra, and a completely distributive double Stone algebra. The primary contribution lies in fully describing the classes of reflexive relations that endow DM(RS) with these classical algebraic structures, thereby filling a theoretical gap in the structural analysis of non-equivalence-based rough set algebras.
📝 Abstract
We consider various types of algebras defined on the completion DM(RS) of the partially ordered set of rough sets induced by a reflexive relation. We restrict ourselves to the cases in which the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe reflexive relations for which DM(RS) has the same properties as in the case of an equivalence relation: it forms a completely distributive and spatial regular double Stone algebra.
