Hypercontact semilattices

TL;DR

This paper introduces the concept of join semilattices with hypercontact relations, providing representation theorems and exploring connections to Boolean algebras, event structures, and hypergraphs.

Contribution

It unifies contact relations in semilattices with hypercontact relations, extending existing theories and offering choice-free proofs and new representation theorems.

Findings

Representation theorems into Boolean algebras with hypercontact relations

Choice-free proofs for most results

Connections to event structures and hypergraphs

Abstract

Contact Boolean algebras are one of the main algebraic tools in region-based theory of space. T. Ivanova provided strong motivations for the study of merely semilattices with a contact relation. Another significant motivation for considering an even weaker underlying structure comes from event structures with binary conflict in the theory of concurrent systems in computer science. All the above-hinted notions deal with a binary contact relation. Several authors suggested the more general study of $n$-ary ``hypercontact'' relations and noticed that, in general, a hypercontact relation cannot be retrieved from just a binary contact relation. A similar evolution occurred in the study of the just mentioned event structures in computer science. In an effort to unify the above lines of research, in this paper we study join semilattices with a hypercontact relation. We provide representation theorems into Boolean algebras, with or without overlap hypercontact relation. With a single exception, our proofs are choice-free. We also present several examples and problems; in particular, we briefly discuss some connections with event structures and hypergraphs.