Publications
In Preparation

M. Abbadini, V. Marra, L. Spada.
Duality for metrically complete latticegroups.
Slides: XXVII Incontro di Logica (September 2022), PhDs in Logic (April 2019).
Poster: UMI Conference (in Italian) (September 2019). 
M. Abbadini, G. Bezhanishvili, L. Carai.
Ideal and MacNeille completion of subordination algebras.
Preprints

M. Abbadini, L. Reggio.
Regular categories and soft sheaf representations.
ArXiv preprint.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 regular categories, thus encompassing a number of ``nonalgebraic'' examples. Our approach is based on the notion of Ksheaf: intuitively, whereas sheaves are defined on open subsets, Ksheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain pointfree 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 a pointfree sheaf representation of Gelfand rings first established by Banaschewski and Vermeulen (Quaest.\ Math., 2011).

M. Abbadini, G. Bezhanishvili, L. Carai.
A generalization of de Vries duality to closed relations between compact Hausdorff spaces.
ArXiv preprint.Abstract
Halmos duality generalizes Stone duality to the category of Stone spaces and continuous relations. This further generalizes to an equivalence (as well as to a dual equivalence) between the category Stone^{R} of Stone spaces and closed relations and the category BA^{S} of boolean algebras and subordination relations. We apply the Karoubi envelope construction to this equivalence to obtain that the category KHausR of compact Hausdorff spaces and closed relations is equivalent to the category DeVS of de Vries algebras and compatible subordination relations. This resolves a problem recently raised in the literature.
We prove that the equivalence between KHaus^{R} and DeV^{S} further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory DeV^{F} of DeV^{S} whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of our approach is that composition of morphisms is usual relation composition.
Papers in PeerReviewed Journals and Proceedings

M. Abbadini, P. Jipsen, T. Kroupa, S. Vannucci.
A finite axiomatization of positive MValgebras.
Algebra Universalis, 83, 28 (2022).
Article, ArXiv preprint.Abstract
Positive MValgebras are the subreducts of MValgebras with respect to the signature {⊕, ⊙, ∨, ∧, 0, 1}. We provide a finite quasiequational axiomatization for the class of such algebras.

M. Abbadini, L. Spada.
Are locally finite MValgebras a variety?
Journal of Pure and Applied Algebra , 226, 4 (2021).
Article, ArXiv preprint.Abstract
We answer Mundici’s problem number 3 (D. Mundici, Advanced Łukasiewicz Calculus and MVAlgebras, Trends in Logic—Studia Logica Library, vol. 35, Springer, Dordrecht (2011), p. 235): Is the category of locally finite MValgebras equivalent to an equational class? We prove:
 The category of locally finite MValgebras is not equivalent to any finitary variety.
 More is true: the category of locally finite MValgebras is not equivalent to any finitelysorted finitary quasivariety.
 The category of locally finite MValgebras is equivalent to an infinitary variety; with operations of at most countable arity.
 The category of locally finite MValgebras is equivalent to a countablysorted finitary variety.
Recordings: Nonclassical Logic Webinar (May 2021). 
M. Abbadini.
Equivalence à la Mundici for commutative latticeordered monoids.
Algebra Universalis, 82, 45 (2021).
Article (open access).Abstract
We provide a generalization of Mundici's equivalence between unital Abelian latticeordered groups and MValgebras: the category of unital commutative latticeordered monoids is equivalent to the category of MVmonoidal algebras. Roughly speaking, unital commutative latticeordered monoids are unital Abelian latticeordered 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, MVmonoidal algebras are MValgebras without the negation x ↦¬x. The primitive operations are ⊕, ⊙, ∨, ∧, 0, 1. A motivating example of MVmonoidal algebra is the negationfree reduct of the standard MValgebra [0,1] ⊆ ℝ. We obtain the original Mundici's equivalence as a corollary of our main result.

M. Abbadini, L. Reggio.
On the axiomatisability of the dual of compact ordered spaces.
Applied Categororical Structures, 28(6):921934 (2020).
Article (open access).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 ℵ_{1}ary variety of algebras. Further, we show that ℵ_{1} is a sharp bound: compact ordered spaces are not dually equivalent to any SPclass of finitary algebras.

M. Abbadini.
Operations that preserve integrability, and truncated Riesz spaces.
Forum Mathematicum, 32(6):14871513.
Article (open access).Abstract
For any real number p ∈ [1, +∞), we characterise the operations ℝ^{I} → ℝ that preserve pintegrability, i.e., the operations under which, for every measure μ, the set L_{p}(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 σcomplete 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 σcomplete truncated Riesz spaces. Analogous results are obtained for operations that preserve pintegrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σcomplete Riesz spaces with weak unit, ℝ is proved to generate this variety, and a concrete model of the free Dedekind σcomplete Riesz spaces with weak unit is exhibited.

M. Abbadini.
Dedekind σcomplete ℓgroups and Riesz spaces as varieties.
Positivity, 24(4):10811100 (2020).
Article, ArXiv preprint.Abstract
We prove that the category of Dedekind σcomplete 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 σcomplete 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 quasivariety. Analogous results are established for the categories of (i) Dedekind σcomplete Riesz spaces with a weak order unit, (ii) Dedekind σcomplete latticeordered groups, and (iii) Dedekind σcomplete latticeordered groups with a weak order unit.

M. Abbadini, F. Di Stefano, L. Spada.
Unification in Łukasiewicz logic with a finite number of variables.
Information Processing and Management of Uncertainty in KnowledgeBased Systems, 1239:622633. Springer International Publishing (2020).
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 MValgebras and rational polyhedra, and a homotopytheoretic argument.

M. Abbadini.
The dual of compact ordered spaces is a variety.
Theory and Applications of Categories, 34(44):14011439 (2019).
Article (open access).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 quasivariety—not finitary, but bounded by ℵ_{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].
2022
2021
2020
2019
PhD thesis

M. Abbadini. Supervisor: Prof. V. Marra.
On the axiomatisability of the dual of compact ordered spaces.
Department of Mathematics of the University of Milan, Italy, 2021.
Thesis (open access).
Video abstract (3 min 38 s).Abstract
We prove that the category of Nachbin’s compact ordered spaces and orderpreserving 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 firstorder 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 latticeordered monoids is equivalent to the category of what we call MVmonoidal algebras. Our proof is independent of Mundici’s theorem.