On the Axiomatisability of the Dual of Compact Ordered Spaces
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On the Axiomatisability of the Dual of Compact Ordered Spaces Marco Abbadini1 · Luca Reggio2 Received: 23 September 2019 / Accepted: 30 July 2020 © The Author(s) 2020
Abstract We provide a direct and elementary proof of the fact that the category of Nachbin’s compact ordered spaces is dually equivalent to an ℵ1 -ary variety of algebras. Further, we show that ℵ1 is a sharp bound: compact ordered spaces are not dually equivalent to any SP-class of finitary algebras. Keywords Compact ordered spaces · Duality · Axiomatisability · Infinitary varieties In 1936, in his landmark paper [21], M. H. Stone described what is nowadays known as Stone duality for Boolean algebras. In modern terms, it states that the category of Boolean algebras with homomorphisms is dually equivalent to the category of totally disconnected compact Hausdorff spaces with continuous maps. If we drop the assumption of total disconnectedness, we are left with the category KH of compact Hausdorff spaces and continuous maps. Duskin showed in 1969 that the opposite category KHop —which, by Gelfand-Naimark duality [10], can be identified with the category of commutative unital C∗ -algebras—is monadic over the category of sets and functions [8, 5.15.3]. In fact, KHop is equivalent to a variety of algebras. Although not finitary, this is an ℵ1 -ary variety. That is, it can be described by operations of at most countably infinite arity. A generating set of operations was exhibited by Isbell [13], while a finite axiomatisation of this variety was provided in [16]. Therefore, if we allow for infinitary operations, Stone duality for Boolean algebras can be lifted to compact Hausdorff spaces, retaining the algebraic nature. Shortly after his paper on the duality for Boolean algebras, Stone published a generalisation of this theory to distributive lattices [23]. In his formulation, the dual category consists of the nowadays called spectral spaces and perfect maps. While spectral spaces are in general not Hausdorff, H. A. Priestley showed in 1970 that they can be equivalently described as certain compact Hausdorff spaces equipped with a partial order relation [19]. More precisely,
Communicated by M. M. Clementino.
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Luca Reggio [email protected]
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Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy
2
Department of Computer Science, University of Oxford, Oxford, UK
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M. Abbadini , L. Reggio
Priestley duality states that the category of (bounded) distributive lattices is dually equivalent to the full subcategory of Nachbin’s compact ordered spaces on the totally order-disconnected objects (cf. Definitions 3 and 19). As with Boolean algebras, one may ask if Priestley duality can be lifted to the category KH≤ of compact ordered spaces while retaining its algebraic op nature. In [12], the authors showed that KH≤ is equivalent to an ℵ1 -ary quasi-variety and partially described its algebraic theory. In the recent work [1], the first-named author proved op that KH≤ is in fact equiva
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