Reducing concept lattices by means of a weaker notion of congruence

TL;DR

This paper introduces a new, weaker form of congruence called local congruence for reducing concept lattices, applicable to both classical and fuzzy formal concept analysis, aiming to minimize information loss.

Contribution

It proposes a novel local congruence relation for concept lattice reduction, analyzing its properties and providing a reduction procedure suitable for classical and fuzzy frameworks.

Findings

The local congruence relation preserves essential lattice structure.

A reduction procedure based on local congruence is developed.

The method is applicable to both classical and fuzzy formal concept analysis.

Abstract

Attribute and size reductions are key issues in formal concept analysis. In this paper, we consider a special kind of equivalence relation to reduce concept lattices, which will be called local congruence. This equivalence relation is based on the notion of congruence on lattices, with the goal of losing as less information as possible and being suitable for the reduction of concept lattices. We analyze how the equivalence classes obtained from a local congruence can be ordered. Moreover, different properties related to the algebraic structure of the whole set of local congruences are also presented. Finally, a procedure to reduce concept lattices by the new weaker notion of congruence is introduced. This procedure can be applied to the classical and fuzzy formal concept analysis frameworks.