Topological Representation of Double Boolean Algebras

TL;DR

This paper develops a topological framework for representing double Boolean algebras (dBas), establishing dualities and embeddings between algebraic and topological structures, and introduces Stone contexts for finite and Boolean cases.

Contribution

It introduces CTSCR topological spaces for representing dBas, proves categorical dualities, and extends the representation theory to finite and Boolean algebras.

Findings

Clopen object oriented protoconcepts form fully contextual dBas.

Clopen object oriented semiconcepts form pure dBas.

Categorical dualities between algebraic and topological structures are established.

Abstract

In formal concept analysis, the collection of protoconcepts of any context forms a double Boolean algebra (dBa) which is fully contextual. Semiconcepts of a context form a pure dBa. The present article is a study on topological representation results for dBas, and in particular, those for fully contextual and pure dBas. In this work, a context on topological spaces (CTS) is considered, and the focus is on a special kind of CTS in which the relation defining the context as well as the converse of the relation are continuous with respect to the topologies. Such CTS are denoted as CTSCR. It is observed that clopen object oriented protoconcepts of a CTSCR form a fully contextual dBa, while clopen object oriented semiconcepts form a pure dBa. Every dBa is shown to be quasi-embeddable into the dBa of clopen object oriented protoconcepts of a particular CTSCR. The quasi-embedding turns into an embedding in case of a contextual dBa, and into an isomorphism, when the dBa is fully contextual. For pure dBas, one obtains an isomorphism with the algebra of clopen object oriented semiconcepts of the CTSCR. Representation of finite dBas and Boolean algebras are also addressed in the process. Abstraction of properties of this CTSCR leads to the definition of Stone contexts. Stone contexts and CTSCR- homeomorphisms are seen to form a category, denoted Scxt. Furthermore, correspondences are observed between dBa isomorphisms and CTSCR-homeomorphisms. This motivates a study of categorical duality of dBas, constituting the second part of the article. Pure dBas and fully contextual dBas along with dBa isomorphisms form categories, denoted PDBA and FCDBA respectively. It is established that PDBA is equivalent to FCDBA, while FCDBA and PDBA are dually equivalent to Scxt.