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

Journal article

It has been known since the work of Duskin and Pelletier four decades ago that Kop, the opposite of the category of compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that Kop 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 Kop. 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.

Authors: Marra, V; Reggio, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 307
• Pages: 253-287
• Publication date: 2016-11-24
• Acceptance date: 2016-11-14
• DOI: 10.1016/j.aim.2016.11.012
• ISSN: 0001-8708
• Language: English
• Keywords: stone duality; lattice-ordered Abelian groups; algebraic theories; FFR; C*-algebras; MV-algebras; axiomatisability; Stone-Weierstrass theorem; rings of continuous functions; Boolean algebras; compact Hausdorff spaces