Continuous Domains in Formal Concept Analysis

TL;DR

This paper introduces a new way to represent continuous domains within Formal Concept Analysis, establishing a categorical framework that connects algebraic structures with formal contexts.

Contribution

It defines attribute continuous formal contexts and concepts, showing they form continuous domains and relate to existing domain subclasses through categorical equivalences.

Findings

Continuous formal concepts form continuous domains.

Every continuous domain can be represented as a continuous formal concept.

The framework includes representations of algebraic and bounded complete domains.

Abstract

Formal Concept Analysis has proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notions of attribute continuous formal context and continuous formal concept are introduced by considering a selection F of fnite subsets of attributes. Our decision of a selection F relies on a kind of generalized interior operators. It is shown that the set of continuous formal concepts forms a continuous domain, and every continuous domain can be obtained in this way. Moreover, an notion of F-morphism is also identified to produce a category equivalent to that of continuous domains with Scott-continuous functions. This paper also consider the representations of various subclasses of continuous domains such as algebraic domains, bounded complete domains and stably continuous semilattices. These results explore the fundamental idea of domain theory in Formal Concept Analysis from a categorical viewpoint.