Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypercontact semilattices

Paolo Lipparini

TL;DR

Hypercontact semilattices extend binary contact relations to $n$-ary compatibility on join semilattices, unifying region-based space theories with concurrent system concepts. The authors develop precise definitions, prove representation theorems showing embeddability into Boolean algebras with overlap or nonoverlap hypercontact and into distributive lattices with additive hypercontact, and establish several equivalent reformulations (topological representations, pre-closures). They also compare these results with the binary case, show non-finite axiomatizability, and explore connections to event structures, graphs, and hypergraphs, offering a broad, choice-light framework for higher-arity contact relations. The work thus provides a robust structural bridge between algebraic, topological, and combinatorial models of compatibility, with potential applications in region-based geometry and concurrency theory.

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.

Hypercontact semilattices

TL;DR

Hypercontact semilattices extend binary contact relations to -ary compatibility on join semilattices, unifying region-based space theories with concurrent system concepts. The authors develop precise definitions, prove representation theorems showing embeddability into Boolean algebras with overlap or nonoverlap hypercontact and into distributive lattices with additive hypercontact, and establish several equivalent reformulations (topological representations, pre-closures). They also compare these results with the binary case, show non-finite axiomatizability, and explore connections to event structures, graphs, and hypergraphs, offering a broad, choice-light framework for higher-arity contact relations. The work thus provides a robust structural bridge between algebraic, topological, and combinatorial models of compatibility, with potential applications in region-based geometry and concurrency theory.

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 -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.
Paper Structure (16 sections, 9 theorems, 15 equations)

This paper contains 16 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. From contact algebras to contact semilattices and posets
  3. From binary contact to $n$-ary hypercontact
  4. Representations of hypercontact semilattices
  5. Some problems
  6. Hypercontact semilattices
  7. Some auxiliary notions and lemmas
  8. Embeddings into overlap Boolean algebras
  9. Embeddings into nonoverlap Boolean algebras
  10. Problems and further remarks
  11. A comment about the axiom of choice
  12. Problems and results about other kinds of representations
  13. Failure of finite axiomatizability
  14. Connections with event structures
  15. Connections with graphs and hypergraphs
  16. ...and 1 more sections

Key Result

Lemma 3.1

Suppose that $\mathbf S$ is a hypercontact semilattice and $\mathbf S$ satisfies m1. Then $\mathbf S$ satisfies the following condition.

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Definition 4.1
  • Theorem 4.2
  • Remark 4.3
  • ...and 12 more