A Categorical View on Algebraic Lattices in Formal Concept Analysis

TL;DR

This paper explores algebraic lattices within formal concept analysis using category theory, connecting concepts like approximation, representation, and duality to deepen understanding of their structure and applications.

Contribution

It introduces a categorical approach to algebraicity in formal concept analysis and links it to Stone duality and existing formal frameworks.

Findings

Equivalence between approximable concepts and algebraic lattices

Representation of algebraic lattices via Stone duality and Scott information systems

Integration of formal concept analysis with domain theory and topology

Abstract

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.