Bohr compactifications of algebras and structures

TL;DR

This paper unifies various categorical compactification constructions, especially Bohr compactifications, by relating natural extensions and topological reflections across different algebraic and ordered structures.

Contribution

It extends the notion of natural extension functors to structures, showing their relation to Bohr compactifications and providing conditions for their coincidence.

Findings

Natural extension and Bohr compactification coincide in zero-dimensional cases.

In ordered structures, the functors often differ but can agree on all objects.

Explicit descriptions of natural extensions are provided for classes with finite, dualisable generators.

Abstract

This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same, in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes---often varieties---are plentiful and varied, and in many cases the associated topological prevariety is standard.