Contact join-semilattices

TL;DR

This paper introduces contact join-semilattices (CJS) and distributive contact join-semilattices (DCJS), generalizing contact algebra by removing certain operations, and provides their set-theoretical and relational representations along with topological and decidability results.

Contribution

It develops new algebraic structures called CJS and DCJS, with representation theorems and decidability results, extending contact algebra in a novel way.

Findings

Set-theoretical representation theorem for CJS

Relational representation theorem for DCJS

Decidability of the universal theory of CJS and DCJS

Abstract

Contact algebra is one of the main tools in region-based theory of space. In \cite{dmvw1, dmvw2,iv,i1} it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices (CJS) and structures, called distributive contact join-semilattices (DCJS). We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal theory of CJS and of DCJS is the same and is decidable.