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

It has been known since the work of Duskin and Pelletier four decades ago that K^op, the opposite of the category of compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that K^op is equivalent to a possibly infinitary variety of algebras Δ in the sense of Słomiński and Linton. Isbell showed in 1982 that the Lawvere–Linton algebraic theory of Δ can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosický independently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of K^op. In particular, Δ is not a finitary variety – Isbell's result is best possible. The problem of axiomatising Δ 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 Δ.