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On the Axiomatisability of the Dual of Compact Ordered Spaces

Published online by Cambridge University Press:  28 February 2022

Marco Abbadini*
Affiliation:
Università degli studi di Milano, Italy, 2021
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Abstract

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (i) any finitely accessible category, (ii) any first-order definable class of structures, and (iii) any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.

Abstract taken directly from the thesis.

E-mail: [email protected]

URL: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf

Type
Thesis Abstracts
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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Footnotes

Supervised by Vincenzo Marra.