Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)

TL;DR

This paper provides a finite axiomatisation of the algebraic theory of the category of compact Hausdorff spaces, using MV-algebras and insights into infinitary operations, advancing the understanding of its algebraic structure.

Contribution

It introduces a finite axiomatisation of the algebraic theory of KH^op, resolving a long-standing open problem in the field.

Findings

Finite axiomatisation of the algebraic theory V

Use of MV-algebras as a key tool

Overcomes previous negative results on axiomatisability

Abstract

It has been known since the work of Duskin and Pelletier four decades ago that KH^op, the category opposite to compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that KH^op is equivalent to a possibly infinitary variety of algebras V in the sense of Slominski and Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosicky independently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of KH^op. In particular, V is not a finitary variety--Isbell's result is best possible. The problem of axiomatising V by equations has remained open. Using the theory of Chang's MV-algebras as a key tool, along with Isbell's fundamental insight on the semantic nature of the infinitary operation, we provide a finite axiomatisation of V.