On the axiomatisability of the dual of compact ordered spaces

TL;DR

This paper establishes a finite axiomatization for the dual category of compact ordered spaces, showing it is equivalent to a variety of algebras with countable operations, and proves limitations on such dualities.

Contribution

It provides the first finite equational axiomatisation of the dual of compact ordered spaces and demonstrates the optimality of the countable arity bound.

Findings

The dual category is equivalent to a variety of algebras with countable arity.

The category of compact ordered spaces is not dually equivalent to any finitary algebraic class.

A finite equational axiomatisation of the dual category is explicitly constructed.

Abstract

We prove that the category of Nachbin's compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we show that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (i) any finitely accessible category, (ii) any first-order definable class of structures, (iii) any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by D. Mundici: our result asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras. Our proof is independent of Mundici's theorem.