Rough sets determined by tolerances

TL;DR

This paper investigates the algebraic structures of rough sets determined by tolerances, showing they form ortholattices and characterizing conditions for complete distributivity and lattice properties, with implications for algebraic and logical frameworks.

Contribution

It characterizes when rough set lattices are ortholattices, Boolean, or complete, and introduces new representations and conditions for their algebraic structure based on tolerances.

Findings

Rough sets form ortholattices for any tolerance R.

Complete distributivity occurs iff R is induced by an irredundant covering.

Disjoint and concept-based representations are Dedekind--MacNeille completions.

Abstract

We show that for any tolerance $R$ on $U$, the ordered sets of lower and upper rough approximations determined by $R$ form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if $R$ is induced by an irredundant covering of $U$, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set $\mathit{RS}$ of rough sets determined by a tolerance $R$ on $U$ is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that $R$ is a tolerance induced by an irredundant covering of $U$ if and only if $\mathit{RS}$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $\mathit{RS}$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set $\mathit{RS}_X$ is a lattice for all $X \subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and $\mathit{RS}_X$ is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind--MacNeille completions of $\mathit{RS}$.