The structure of rough sets defined by reflexive relations

TL;DR

This paper explores the order-theoretic properties of Dedekind-MacNeille completions of rough set systems induced by reflexive relations, revealing conditions under which they form Nelson algebras and characterizing their structure.

Contribution

It provides a detailed analysis of the structure of DM(RS) for reflexive relations, including conditions for Nelson algebra formation and the introduction of the core of a relational neighborhood.

Findings

DM(RS) can form a Nelson algebra even for non-transitive reflexive relations

Characterization of when DM(RS) is a spatial completely distributive lattice

Introduction of the core of a relational neighborhood as a key concept

Abstract

For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.