Boolean Substructures in Formal Concept Analysis

TL;DR

This paper explores the relationship between Boolean subcontexts and Boolean suborders in formal concept analysis, introducing mappings and structures to better understand their interplay within finite contexts and lattices.

Contribution

It defines new mappings between subcontexts and suborders, and introduces closed-subcontexts to analyze sublattices in formal concept analysis.

Findings

Established correspondence between Boolean subcontexts and suborders.

Introduced mappings and structural properties linking contexts and lattices.

Extended the concept of closed relations to closed-subcontexts.

Abstract

It is known that a (concept) lattice contains an n-dimensional Boolean suborder if and only if the context contains an n-dimensional contra-nominal scale as subcontext. In this work, we investigate more closely the interplay between the Boolean subcontexts of a given finite context and the Boolean suborders of its concept lattice. To this end, we define mappings from the set of subcontexts of a context to the set of suborders of its concept lattice and vice versa and study their structural properties. In addition, we introduce closed-subcontexts as an extension of closed relations to investigate the set of all sublattices of a given lattice.