On the axiomatisability of the dual of compact ordered spaces
Marco Abbadini
TL;DR
The thesis proves that the opposite category of Nachbin’s compact ordered spaces, CompOrd^{op}, is (i) dually equivalent to a variety of algebras with infinitary operations of at most countable arity, and (ii) admits a finite equational axiomatisation via limit 2-divisible MV-monoidal algebras, extending Mundici-type correspondences. The core technical advance is showing that equivalence corelations in CompOrd^{op} are effective, enabling a typical variety-theoretic presentation. It further establishes strong negative results: CompOrd^{op} cannot be dually equivalent to any finitary algebra variety, nor to any finitely accessible category, nor to any first-order definable class, or to any SP-class of finitary algebras. To achieve the finite axiomatisation, the work introduces MV-monoidal algebras as unit intervals of unital commutative distributive ℓ-monoids and proves their categorical equivalence with these interval structures, mirroring Mundici’s classical result for MV-algebras and lattice-ordered groups. Collectively, the results illuminate how ordered topological dualities generalise Stone and Priestley dualities to a robust infinitary algebraic setting, with implications for logical semantics and ordered topological representations and offering a concrete, finite axiomatisation in the finite-syntactic sense through MV-monoidal geometry.
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 show 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, (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 D. Mundici: our result asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras. Our proof is independent of Mundici's theorem.