12 research outputs found

    Dedekind sigma-complete l-groups and Riesz spaces as varieties

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    We prove that the category of Dedekind σ\sigma-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that R\mathbb{R}, regarded as a Dedekind σ\sigma-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind σ\sigma-complete Riesz spaces with a weak order unit, (ii) Dedekind σ\sigma-complete lattice-ordered groups, and (iii) Dedekind σ\sigma-complete lattice-ordered groups with a weak order unit.Comment: 15 page

    The dual of compact ordered spaces is a variety

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    In a recent paper (2018), D. Hofmann, R. Neves and P. Nora proved that the dual of the category of compact partially ordered spaces and monotone continuous maps is a quasi-variety - not finitary, but bounded by 1\aleph_1. An open question was: is it also a variety? We show that the answer is affirmative. We describe the variety by means of a set of finitary operations, together with an operation of countably infinite arity, and equational axioms. The dual equivalence is induced by the dualizing object [0,1]

    Operations that preserve integrability, and truncated Riesz spaces

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    For any real number p[1,+)p\in [1,+\infty), we characterise the operations RIR\mathbb{R}^I \to \mathbb{R} that preserve pp-integrability, i.e., the operations under which, for every measure μ\mu, the set Lp(μ)\mathcal{L}^p(\mu) is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind σ\sigma-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that R\mathbb{R} generates this variety. From this, we exhibit a concrete model of the free Dedekind σ\sigma-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve pp-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σ\sigma-complete Riesz spaces with weak unit, R\mathbb{R} is proved to generate this variety, and a concrete model of the free Dedekind σ\sigma-complete Riesz spaces with weak unit is exhibited.Comment: Changed the definition of "conditionally partitionable measure space", results unchanged; minor change

    Barr-Exact Categories and Soft Sheaf Representations

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    It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness. In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of "non-algebraic" examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames. These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).Comment: 39 pages. v3: presentation improved. Title modified to reflect change in presentatio

    On the axiomatisability of the dual of compact ordered spaces

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    We provide a direct and elementary proof of the fact that the category of Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered spaces are not dually equivalent to any SP-class of finitary algebras.Comment: 10 pages. v3: minor changes. To appear in Applied Categorical Structure

    Stone-Gelfand duality for metrically complete lattice-ordered groups

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    We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing 11, namely, either 1nZ\frac{1}{n}\mathbb{Z} for an integer n=1,2,n = 1, 2, \dots, or the whole of R\mathbb{R}. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic" compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces whose each point is assigned the entire group of real numbers. In the introduction we indicate motivations from and connections with the theory of dimension groups.Comment: 24 pages, 2 figure

    Ideal and MacNeille completions of subordination algebras

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    S5\mathsf{S5}-subordination algebras were recently introduced as a generalization of de Vries algebras, and it was proved that the category SubS5S\mathsf{SubS5^S} of S5\mathsf{S5}-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of S5\mathsf{S5}-subordination algebras, and utilize the relational nature of the morphisms in SubS5S\mathsf{SubS5^S} to prove that the MacNeille completion functor establishes an equivalence between SubS5S\mathsf{SubS5^S} and its full subcategory consisting of de Vries algebras. We also generalize ideal completions of boolean algebras to the setting of S5\mathsf{S5}-subordination algebras and prove that the ideal completion functor establishes a dual equivalence between SubS5S\mathsf{SubS5^S} and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of SubS5S\mathsf{SubS5^S} corresponding to continuous relations and continuous functions between compact Hausdorff spaces

    Vietoris endofunctor for closed relations and its de Vries dual

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    We generalize the classic Vietoris endofunctor to the category of compact Hausdorff spaces and closed relations. The lift of a closed relation is done by generalizing the construction of the Egli-Milner order. We describe the dual endofunctor on the category of de Vries algebras and subordinations. This is done in several steps, by first generalizing the construction of Venema and Vosmaer to the category of boolean algebras and subordinations, then lifting it up to S5\mathsf{S5}-subordination algebras, and finally using MacNeille completions to further lift it to de Vries algebras. Among other things, this yields a generalization of Johnstone's pointfree construction of the Vietoris endofunctor to the category of compact regular frames and preframe homomorphisms

    A generalization of de Vries duality to closed relations between compact Hausdorff spaces

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    Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category De VS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to De VS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and De VS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De VF of De VS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition

    A Finite Axiomatization of Positive MV-Algebras

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    Positive MV-algebras are the subreducts of MV-algebras with respect to the signature {⊕,⊙,∨,∧,0,1}. We provide a finite quasi-equational axiomatization for the class of such algebras
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