Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The structure of rough sets defined by reflexive relations

Jouni Järvinen, Sándor Radeleczki

TL;DR

The paper analyzes rough-set systems built from reflexive binary relations through their Dedekind–MacNeille completion DM(RS). It provides precise descriptions of completely join-irreducible elements and atoms, derives criteria for complete distributivity and spatiality via the underlying approximation lattices, and establishes when DM(RS) forms a Nelson algebra using a core-based interpolation condition. Notably, DM(RS) can be a Nelson algebra even without transitivity, extending classical results known for quasiorders and equivalences. These findings broaden the algebraic and logical applications of rough-set theory and connect it with Alexandroff topologies and Heyting/quasi-Nelson frameworks, enabling more general deduction methods for relation-based rough sets.

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.

The structure of rough sets defined by reflexive relations

TL;DR

The paper analyzes rough-set systems built from reflexive binary relations through their Dedekind–MacNeille completion DM(RS). It provides precise descriptions of completely join-irreducible elements and atoms, derives criteria for complete distributivity and spatiality via the underlying approximation lattices, and establishes when DM(RS) forms a Nelson algebra using a core-based interpolation condition. Notably, DM(RS) can be a Nelson algebra even without transitivity, extending classical results known for quasiorders and equivalences. These findings broaden the algebraic and logical applications of rough-set theory and connect it with Alexandroff topologies and Heyting/quasi-Nelson frameworks, enabling more general deduction methods for relation-based rough sets.

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.

Paper Structure

This paper contains 6 sections, 29 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rough sets by reflexive relations
  3. Completely join-irreducible elements and atoms
  4. On complete distributivity
  5. Nelson algebras defined on $\mathrm{DM(RS)}$
  6. Some concluding remarks

Key Result

Proposition 2.1

The meets and joins are formed in $\mathrm{DM(RS)}$ as in Eq:Meet and Eq:Join, respectively.

Figures (4)

  • Figure 1: The complete lattice $\mathrm{DM(RS)}$ of Example \ref{['exa:JoinRS']}, forming a regular double Stone algebra isomorphic to $\mathbf{2} \times \mathbf{3}$.
  • Figure 2: The directional similarity relation $R$.
  • Figure 3: The distributive lattices $\wp(U)^\vartriangle$ and $\wp(U)^\blacktriangle$ of Example \ref{['exa:compl_distr']}.
  • Figure 4: The Hasse diagram of $\mathrm{RS} = \mathrm{DM(RS)}$. Its completely join-irreducible elements are marked with filled circles.

Theorems & Definitions (62)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 3.1
  • ...and 52 more