Publications

    2025

  1. Stone-Gelfand duality for metrically complete lattice-ordered groups.
    Advances in Mathematics, 461:110067 (2025).
    M. Abbadini, V. Marra, L. Spada.
    Article.
    Abstract

    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 1⁄n ℤ for an integer n = 1, 2, ..., or the whole of 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.

    Recording: ASL North American Meeting (May 2024).
    Slides: ASL North American Meeting (May 2024), Seminar at Laboratoire Méthodes Formelles (Jan. 2023), XXVII Incontro di Logica (Sept. 2022), PhDs in Logic (April 2019).
    Poster: UMI Conference (in Italian) (Sept. 2019).

    2. 2024

  2. Vietoris endofunctor for closed relations and its de Vries dual.
    Topology Proceedings, 64:213-250 (2024).
    M. Abbadini, G. Bezhanishvili, L. Carai.
    Article, ArXiv preprint.
    Abstract

    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-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.

    Recordings: AILA Meeting (Sept 2024), TACL (July 2024), SumTopo (July 2024).
    Slides: SumTopo 2024 (July 2024), TACL 2024 (July 2024).

  3. Duality for coalgebras for Vietoris and monadicity.
    Journal of Symbolic Logic, to appear.
    M. Abbadini, I. Di Liberti.
    Article.
    Abstract

    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 Jósson-Tarski duality for modal algebras beyond the 0-dimensional setting.

    Recordings: Logic Colloquium 2023 (July 2023), 3rd Itaca Workshop (Dec 2022).
    Slides: Logic Colloquium (June 2023), 3rd Itaca Workshop (Dec. 2022).

  4. MacNeille completions of subordination algebras.
    Cahiers de Topologie et Géométrie Différentielle Catégoriques, 65(2):151-199 (2024).
    M. Abbadini, G. Bezhanishvili, L. Carai.
    Article.
    Abstract

    S5-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category SubS5 of 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-subordi\-nation algebras, and utilize the relational nature of the morphisms in SubS5 to prove that the MacNeille completion functor establishes an equivalence between SubS5 and its full subcategory consisting of de Vries algebras. We also show that the round ideal functor establishes a dual equivalence between SubS5 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 SubS5 corresponding to continuous relations and continuous functions between compact Hausdorff spaces.


    5. 2023

  5. A generalization of de Vries duality to closed relations between compact Hausdorff spaces.
    Topology and its Applications, 337:108641 (2023).
    M. Abbadini, G. Bezhanishvili, L. Carai.
    Article.
    Abstract

    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 DeVS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to DeVS, thus resolving a problem recently raised in the literature.

    The equivalence between KHausR and DeVS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory DeVF of DeVS 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.

    Slides: Seminar at IRIF (Nov. 2022), TACL 2022 (June 2022), LLAMA Seminar (May 2022).

  6. Barr-exact categories and soft sheaf representations.
    Journal of Pure and Applied Algebra, 227(12):107413 (2023).
    M. Abbadini, L. Reggio.
    Article.
    Abstract

    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 the 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 obtain 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).

    Slides: Seminar at the Université catholique de Louvain (Nov. 2023), PSSL 108 (Sept. 2023).

    7. 2022

  7. A finite axiomatization of positive MV-algebras.
    Algebra Universalis, 83, 28 (2022).
    M. Abbadini, P. Jipsen, T. Kroupa, S. Vannucci.
    Article, ArXiv preprint.
    Abstract

    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.

    Recordings: Algebra Week 2023 (July 2023), BLAST 2023 (May 2023).
    Slides: Third Algebra Week (July 2023), BLAST 2023 (May 2023), Seminar at the University of Bern (April 2023).

  8. Are locally finite MV-algebras a variety?
    Journal of Pure and Applied Algebra , 226, 4 (2022).
    M. Abbadini, L. Spada.
    Article, ArXiv preprint.
    Abstract

    We answer Mundici’s problem number 3 (D. Mundici, Advanced Łukasiewicz Calculus and MV-Algebras, Trends in Logic—Studia Logica Library, vol. 35, Springer, Dordrecht (2011), p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove:

    1. The category of locally finite MV-algebras is not equivalent to any finitary variety.
    2. More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety.
    3. The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity.
    4. 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 known categorical characterisations of varieties and quasi-varieties. In fact, no knowledge of MV-algebras is needed, apart from the aforementioned duality.

    Recording: Nonclassical Logic Webinar (May 2021).
    Slides: Nonclassical Logic Webinar (May 2021).

    9. 2021

  9. Equivalence à la Mundici for commutative lattice-ordered monoids.
    Algebra Universalis, 82, 45 (2021).
    M. Abbadini.
    Article.
    Abstract

    We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x ↦-x. The primitive operations are +, ∨, ∧, 0, 1, -1. A prime example of these structures is ℝ, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x ↦¬x. The primitive operations are ⊕, ⊙, ∨, ∧, 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0,1] ⊆ ℝ. We obtain the original Mundici's equivalence as a corollary of our main result.

    Recordings: Algebra Week 2023 (July 2023), BLAST 2023.
    Slides: DOCToR (July 2021), Nonclassical Logic Seminar (May 2020), University of Salerno (Feb. 2020).

    10. 2020

  10. On the axiomatisability of the dual of compact ordered spaces.
    Applied Categorical Structures, 28(6):921-934 (2020).
    M. Abbadini, L. Reggio.
    Article.
    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.

    Recording: Category Theory 20->21 (Sept. 2021).
    Slides: Category Theory 20->21 (Sept. 2021), BLAST (June 2021), Ph.D. defense (April 2021), AAA99 (Feb. 2020), Czech Academy of Sciences (Feb. 2019).

  11. Operations that preserve integrability, and truncated Riesz spaces.
    Forum Mathematicum, 32(6):1487-1513 (2020).
    M. Abbadini.
    Article, ArXiv preprint.
    Abstract

    For any real number p ∈ [1, +∞), we characterise the operations ℝI → ℝ that preserve p-integrability, i.e., the operations under which, for every measure μ, the set Lp(μ) 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-complete truncated Riesz spaces, where truncation is meant in the sense of R.N. Ball. We also prove that ℝ generates this variety. From this, we exhibit a concrete model of the free Dedekind σ-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 σ-complete Riesz spaces with weak unit, ℝ is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited.

    Slides: BLAST (Aug. 2018).

  12. Unification in Łukasiewicz logic with a finite number of variables.
    Information Processing and Management of Uncertainty in Knowledge-Based Systems, 1239:622-633. Springer International Publishing (2020).
    M. Abbadini, F. Di Stefano, L. Spada.
    Article.
    Abstract

    We prove that the unification type of Łukasiewicz logic with a finite number of variables is either infinitary or nullary. To achieve this result we use Ghilardi’s categorical characterisation of unification types in terms of projective objects, the categorical duality between finitely presented MV-algebras and rational polyhedra, and a homotopy-theoretic argument.


  13. Dedekind σ-complete ℓ-groups and Riesz spaces as varieties.
    Positivity, 24(4):1081-1100 (2020).
    M. Abbadini.
    Article.
    Abstract

    We prove that the category of Dedekind σ-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, regarded as a Dedekind σ-complete Riesz space, generates this category as a variety; further, we use this fact to obtain the even stronger result that R generates this category as a quasi-variety. Analogous results are established for the categories of (i) Dedekind σ-complete Riesz spaces with a weak order unit, (ii) Dedekind σ-complete lattice-ordered groups, and (iii) Dedekind σ-complete lattice-ordered groups with a weak order unit.


    14. 2019

  14. The dual of compact ordered spaces is a variety.
    Theory and Applications of Categories, 34(44):1401-1439 (2019).
    M. Abbadini.
    Article.
    Abstract

    In a recent paper (2018), D. Hofmann, R. Neves and P. Nora proved that the dual of the category of compact ordered spaces and monotone continuous maps is a quasi-variety—not finitary, but bounded by ℵ1. 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].

    Recording: Category Theory 20->21 (Sept. 2021).
    Slides: Category Theory 20->21 (Sept. 2021), BLAST (June 2021), Ph.D. defense (April 2021), AAA99 (Feb. 2020), Czech Academy of Sciences (Feb. 2019).

Preprints

  • Freely adding one layer of quantifiers to a Boolean doctrine.
    M. Abbadini, F. Guffanti.
    ArXiv preprint.
    Abstract We show how to freely add the first layer of quantifier alternation depth to a Boolean doctrine over a small base category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. To achieve this, we characterize, within the doctrinal setting, the classes A of quantifier-free formulas for which there is a model M such that A is precisely the class of formulas whose universal closure is valid in M.
    Slides: Seminar at the University of Milan (Oct 2024), Seminar at the University of Manchester (Oct 2024), Seminar at the University of Padua (May 2024).

  • Barr-coexactness for metric compact Hausdorff spaces.
    M. Abbadini, D. Hofmann.
    ArXiv preprint.
    Abstract

    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 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.


  • Varieties of MV-monoids and positive MV-algebras.
    M. Abbadini, S. Fioravanti, P. Aglianò.
    ArXiv preprint.
    Abstract

    MV-monoids are algebras ⟨A, ∨, ∧, ⊕, ⊙, 0, 1⟩ where ⟨A, ∨, ∧, 0, 1⟩ is a bounded distributive lattice, both ⟨A, ⊕, 0⟩ and ⟨A, ⊕, 1⟩ are commutative monoids, and some further connecting axioms are satisfied. Every MV-algebra in the signature {⊕, ¬, 0} is term equivalent to an algebra that has an MV-monoid as a reduct, by defining, as standard, 1 := ¬0, x ⊙ y := ¬(¬x ⊕ ¬y), x ∨ y := (x ⊙ ¬y) ⊕ y and x ∧ y := ¬(¬x ∨ ¬y). Particular examples of MV-monoids are positive MV-algebras, i.e. the {∨, ∧, ⊕, ⊙, 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.


  • Quantifier-free formulas and quantifier alternation depth in doctrines.
    M. Abbadini, F. Guffanti.
    ArXiv preprint.
    Abstract

    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.

    Slides: Seminar at the University of Padua (May 2024).

PhD thesis

  • On the axiomatisability of the dual of compact ordered spaces.
    M. Abbadini. Supervisor: Prof. V. Marra.
    Department of Mathematics, University of Milan, Italy (2021).
    Thesis.
    Video abstract (3 min 38 s).
    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.

    (Abstract published in the Bulletin of Symbolic Logic.)