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Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a \emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.

MV-monoids are algebras $\langle A,\vee,\wedge, \oplus,\odot, 0,1\rangle$ where $\langle A, \vee, \wedge, 0, 1\rangle$ is a bounded distributive lattice, both $\langle A, \oplus, 0 \rangle$ and $\langle A, \odot, 1\rangle$ are commutative monoids, and some further connecting axioms are satisfied. Every MV-algebra in the signature $\{\oplus,\neg,0\}$ is term equivalent to an algebra that has an MV-monoid as a reduct, by defining, as standard, $1:= \neg 0$, $x \odot y := \neg(\neg x \oplus\neg y)$, $x \vee y := (x \odot \neg y) \oplus y$ and $x \wedge y := \neg(\neg x \vee \neg y)$. Particular examples of MV-monoids are positive MV-algebras, i.e. the $\{\vee, \wedge, \oplus, \odot, 0, 1\}$-subreducts of MV-algebras. Positive MV-algebras form a peculiar quasivariety in the sense that, albeit having a logical motivation (being the quasivariety of subreducts of MV-algebras), it is not the equivalent quasivariety semantics of any logic. In this paper, we study the lattices of subvarieties of MV-monoids and of positive MV-algebras. In particular, we characterize and axiomatize all almost minimal varieties of MV-monoids, we characterize the finite subdirectly irreducible positive MV-algebras, and we characterize and axiomatize all varieties of positive MV-algebras.

This paper aims at providing a first step towards a doctrinal approach to quantifier-free formulas and quantifier alternation depth modulo a first-order theory. The set of quantifier-free formulas modulo a first-order theory is axiomatized by what we call a quantifier-free fragment of a Boolean doctrine with quantifiers. Rather than being an intrinsic notion, a quantifier-free fragment is an additional structure on a Boolean doctrine with quantifiers. Under a smallness assumption, the structures occurring as quantifier-free fragments of some Boolean doctrines with quantifiers are precisely the Boolean doctrines (without quantifiers). To obtain this characterization, we prove that every Boolean doctrine over a small base category admits a quantifier completion, of which it is a quantifier-free fragment. Furthermore, the stratification by quantifier alternation depth of first-order formulas motivates the introduction of quantifier alternation stratified Boolean doctrines. While quantifier-free fragments are defined in relation to an "ambient" Boolean doctrine with quantifiers, a quantifier alternation stratified Boolean doctrine requires no such ambient doctrine. Indeed, it consists of a sequence of Boolean doctrines (without quantifiers) with connecting axioms. We show such sequences to be in one-to-one correspondence with pairs consisting of a Boolean doctrine with quantifiers and a quantifier-free fragment of it., Comment: Comments are welcome! (This is the second version of a much longer manuscript. For better readability, we extracted and expanded the first part of the previous version. In this version we left out Section 5-6 and both Appendices of the original manuscript, which we plan to ripropose in some new form in the future.)

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 $\mathsf{S5}$-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.

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over Set. We deliver an analogous result for the upper, lower and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jonsson-Tarski duality for modal algebras beyond the 0-dimensional setting., Comment: 29 pages. Comments are welcome!

$\mathsf{S5}$-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-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 $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each $\mathsf{S5}$-subordination algebra the frame of its round ideals establishes a dual equivalence between $\mathsf{SubS5^S}$ 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 $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces.

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 $1$, namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the whole of $\mathbb{R}$. 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 figures

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. v4: revised version, to appear in the Journal of Pure and Applied Algebra

Stone duality generalizes to an equivalence between the categories $\mathsf{Stone}^{\mathsf{R}}$ of Stone spaces and closed relations and $\mathsf{BA}^\mathsf{S}$ of boolean algebras and subordination relations. Splitting equivalences in $\mathsf{Stone}^{\mathsf{R}}$ yields a category that is equivalent to the category $\mathsf{KHaus}^\mathsf{R}$ of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in $\mathsf{BA}^\mathsf{S}$ yields a category that is equivalent to the category $\mathsf{DeV^S}$ of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that $\mathsf{KHaus}^\mathsf{R}$ is equivalent to $\mathsf{DeV^S}$, thus resolving a problem recently raised in the literature. The equivalence between $\mathsf{KHaus}^\mathsf{R}$ and $\mathsf{DeV^S}$ further restricts to an equivalence between the category ${\mathsf{KHaus}}$ of compact Hausdorff spaces and continuous functions and the wide subcategory $\mathsf{DeV^F}$ of $\mathsf{DeV^S}$ 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., Comment: 18 pages

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature $\{\oplus, \odot, \lor, \land, 0, 1\}$. We provide a finite quasi-equational axiomatization for the class of such algebras.

We answer Mundici's problem number 3 (D. Mundici. Advanced {\L}ukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove: (i) The category of locally finite MV-algebras is not equivalent to any finitary variety. (ii) More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety. (iii) The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity. (iv) The category of locally finite MV-algebras is equivalent to a countably-sorted finitary variety. Our proofs rest upon the duality between locally finite MV-algebras and the category of multisets by R. Cignoli, E. J. Dubuc and D. Mundici, and categorical characterisations of varieties and quasi-varieties proved by J. Duskin, J. R. Isbell, F. W. Lawvere and others. In fact no knowledge on MV-algebras is needed, apart from the aforementioned duality.

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 Structures

We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered groups is equivalent to the category of MV-monoidal algebras. Roughly speaking, the structures we call unital commutative lattice-ordered groups are unital Abelian lattice-ordered groups without the unary operation $x \mapsto -x$. The primitive operations are $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example of these structures is $\mathbb{R}$, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $x \mapsto \lnot x$. The primitive operations are $\oplus$, $\odot$, $\lor$, $\land$, $0$, $1$. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $[0, 1] \subseteq \mathbb{R}$. We obtain the original Mundici's equivalence as a corollary of our main result.

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 $\aleph_1$. 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].

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

We prove that the category of Dedekind $\sigma$-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 $\mathbb{R}$, regarded as a Dedekind $\sigma$-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 $\sigma$-complete Riesz spaces with a weak order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii) Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit., Comment: 15 pages