Rough sets determined by tolerances

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 $RS$ 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 $RS$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $RS$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set ${RS}_X$ is a lattice for all $X\subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and ${RS}_X$ 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 $RS$.

arXiv commentRevised version (28 pages, 1 figure)

References in corpus (4)

• Rough Sets Determined by Quasiorders
• Information completeness in Nelson algebras of rough sets induced by quasiorders
• Representation of Nelson Algebras by Rough Sets Determined by Quasiorders
• Representing distributive lattices with Galois connections in terms of rough sets

Cited by in corpus (6)

• Representing regular pseudocomplemented Kleene algebras by tolerance-based rough sets
• Defining rough sets using tolerances compatible with an equivalence
• Algebraic Approach to Directed Rough Sets
• Defining rough sets as core-support pairs of three-valued functions
• The structure of rough sets defined by reflexive relations
• Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets