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.

Stone duality above dimension zero: Axiomatising the algebraic theory of C(X) / V. Marra, L. Reggio. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 307(2017), pp. 253-287. [10.1016/j.aim.2016.11.012]

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

V. Marra
Primo
;
2017

Abstract

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.
Algebraic theories; Axiomatisability; Boolean algebras; C⁎-algebras; Compact Hausdorff spaces; Lattice-ordered Abelian groups; MV-algebras; Rings of continuous functions; Stone duality; Stone–Weierstrass Theorem
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
2017
24-nov-2016
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/460952
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