On the axiomatisability of the dual of compact ordered spaces

TL;DR

This paper proves that the category of compact ordered spaces is dually equivalent to a specific type of algebraic variety, establishing the precise bounds of this equivalence and showing it does not extend to finitary algebras.

Contribution

It provides a direct proof of the duality between compact ordered spaces and an Aleph_1-ary algebraic variety, clarifying the limits of this duality.

Findings

Duality between compact ordered spaces and Aleph_1-ary algebras

Aleph_1 is the sharp bound for this duality

Compact ordered spaces are not dual to any finitary algebra class

Abstract

We provide a direct and elementary proof of the fact that the category of Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered spaces are not dually equivalent to any SP-class of finitary algebras.