Tangled Closure Algebras

TL;DR

This paper develops an algebraic framework for tangled closure operations in topology, demonstrating the existence of complex algebraic structures with unique embedding properties and limitations in representation.

Contribution

It introduces an abstract algebraic formulation of tangled closure, generalizes existing theories, and constructs examples with novel embedding and completeness properties.

Findings

Dissectable tangled closure algebras contain all finite tangled closure algebras.

Existence of tangled closure algebras that lack MacNeille completions.

Some tangled closure algebras cannot be embedded into complete or spatial algebras.

Abstract

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.