Reducing concept lattices by means of a weaker notion of congruence

TL;DR

In this paper, a special kind of equivalence relation to reduce concept lattices is considered, which will be called local congruence, based on the notion of congruent on lattices, with the goal of losing as less information as possible and being suitable for the reduction of concept lattice.

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.

Figures (11)

• Figure 1: Opposite sides of a quadrilateral.
• Figure 2: The original concept lattice (left), the obtained reduction in bmrd:RSTFCA:c (middle) and the least congruence containing the previously reduction (right).
• Figure 3: Example of $\delta$-sequence.
• Figure 4: Example of $\delta$-cycle.
• Figure 5: Example where $\preceq_\delta$ is not a partial order.

Theorems & Definitions (36)

• definition 1
• definition 2
• definition 3
• Lemma 1: DaveyPriestley
• Theorem 2: DaveyPriestley
• Lemma 3: DaveyPriestley
• definition 4
• definition 5
• Proposition 4: bmrd:RSTFCA:c
• Proposition 5: bmrd:RSTFCA:c