Talks

You can check out some of my presentations in Youtube.

Invited Plenary Talks at International Conferences

Boolean algebras, Lattices, Universal Algebra, Set Theory, Topology 2023. University of North Carolina at Charlotte, NC, USA. (17 May 2023)
Positive MV-algebras.
Recording, Abstract, Slides.

Invited Talks at Special Sessions in International Conferences

2024 Association for Symbolic Logic North American Annual Meeting. Iowa State University, Ames, IA, USA. Invited to the Special Session on Algebraic Logic. (15 May 2024)
A duality for metrically complete lattice-ordered groups.
Recording, Abstract, Slides.

Invited Talks at International Workshops

Third Algebra Week (2023). University of Siena, Italy. (5 July 2023)
Positive subreducts of MV-algebras.
Recording, Abstract, Slides.

Duality, Order, (Co)algebras, Topology, and Related topics, Online. (7 July 2021)
Unit intervals of unital commutative distributive ℓ-monoids.

Abstract

Given a commutative distributive ℓ-monoid (M; ∨, ∧, +, 0) and an invertible element u ≥ 0 in M, we equip the set of elements of M between 0 and u with the MV-flavored operations ∨, ∧, ⊕, ⊙, 0, 1. For the algebras arising in this manner, we provide an axiomatization that is both equational and finite, and we name these algebras MV-monoidal algebras.

From a categorical perspective, we establish an adjunction that restricts to an equivalence between commutative distributive l-monoids with strong order-unit and MV-monoidal algebras. The equivalence can be further restricted to the celebrated equivalence between Abelian ℓ-groups with strong order-unit and MV-algebras.

Slides.

Third Algebra Week (2020). University of Siena, Italy. (June 2020 - Cancelled due to COVID-19 pandemic)
Equivalence à la Mundici for lattice-ordered monoids.

Invited Talks at National Workshops

4^th Southern and Midlands Logic Seminar. University of Birmingham, UK. (13 Dec 2023)
Natural dualities.

Abstract

In 1936 Stone proved that every Boolean algebra is representable as a subalgebra of the Boolean algebra of subsets of some set. In fact, Stone proved much more, namely that the category of Boolean algebras is dually equivalent to the category of Stone spaces. The beauty of this theorem is that it translates algebraic problems, normally stated in abstract symbolic language, into dual, topological problems, where our geometric intuitions can be brought to bear.

A plethora of similar dualities followed, a prime example being Priestley duality for bounded distributive lattices. In the 80s, these dualities were systematised in the theory of natural dualities. I will give an overview of this theory, which is broad enough to encompass many known dualities, yet concrete enough to generate new ones. Moreover, I will mention an ongoing joint work with Adam Přenosil on extending a portion of this theory to the case in which the so-called dualising object is possibly infinite.

Slides.

Contributed Talks at International Conferences

XXVIII Logic Meeting of the Italian Association of Logic and its Applications. Udine, Italy. (4 Sept 2024)
Vietoris endofunctor for closed relations and its de Vries dual.
Recording, Abstract. Slides.

38^th summer conference on topology and its applications. University of Coimbra, Portugal. (9 July 2024)
Vietoris endofunctor for closed relations and its de Vries dual.
Recording, Abstract, Slides.

Topology, Algebra, and Categories in Logic 2024. University of Barcelona, Spain. (4 July 2024)
Vietoris endofunctor for closed relations and its de Vries dual.
Recording, Abstract, Slides.

(104^th Workshop on General Algebra "Arbeitstagung Allgemeine Algebra"). South-West University “Neofit Rilsk”, Blagoevgrad, Bulgaria. (10 Feb 2024)
Natural dualities with an infinite dualizing object.

Abstract

In 1936 Stone proved that every Boolean algebra is representable as a subalgebra of the Boolean algebra of subsets of some set. In fact, Stone proved much more, namely that the category of Boolean algebras is dually equivalent to the category of Stone spaces. The beauty of this theorem is that it translates algebraic problems, normally stated in abstract symbolic language, into dual, topological problems, where our geometric intuitions can be brought to bear.

A plethora of similar dualities for general algebraic structures followed, a prime example being Priestley duality for bounded distributive lattices. In the 80s, these dualities were systematised in the theory of natural dualities. This theory is broad enough to encompass many known dualities, yet concrete enough to generate new ones.

I will present an ongoing joint work with Adam Přenosil on extending a portion of this theory to the case in which the so-called dualizing object is possibly infinite. In particular, given a possibly infinite hereditarily finitely subdirectly irreducible algebra L with a near-unanimity term, we provide a duality for the category of algebras A in ISP(L) such that for each x in A the set {h(a) : h in hom(A, L)} is finite.

Slides.

"Modalities in Substructural Logics: Theory Methods and Applications" Workshop 2023. Vienna, Austria. (29 Sept 2023)
Stone duality for finitely valued algebras with a near-unanimity term.
Abstract, Slides.

108^th Peripatetic Seminar on Sheaves and Logic. University of Palermo, Terrasini, Italy. (17 Sept 2023)
Soft sheaf representations in Barr-exact categories.

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.

Logic, Algebra and Truth Degrees 2023. Ivane Javakhishvili State University, Tbilisi, Georgia. (11 Sept 2023)
Free extensions.
Abstract, Slides.

Logic Colloquium 2023. University of Milan, Italy. (8 June 2023)
Jónsson-Tarski duality beyond dimension 0.
Recording, Abstract, Slides.

3^rd Itaca Workshop, University of Pisa, Italy. (21 Dec 2022)
Comonadicity over Set of coalgebras of Vietoris functors.
Recording, Abstract, Slides.

Topology, Algebra, and Categories in Logic 2022. University of Coimbra, Portugal. (24 June 2022)
An approach à la de Vries for compact Hausdorff spaces and closed relations.
Abstract, Slides.

Category Theory 20->21. University of Genoa, Italy. (1 Sept 2021)
The opposite of the category of compact ordered spaces is monadic over the category of sets .
Recording, Abstract, Slides.

Boolean Algebras, Lattices, Universal Algebra, Set Theory, Topology 2021. New Mexico State University, Las Cruces, USA, online. (13 June 2021)
The opposite of the category of compact ordered spaces as an infinitary variety.
Recording, Abstract, Slides.

99^th Workshop on General Algebra "Arbeitstagung Allgemeine Algebra". University of Siena, Italy. (21 Feb 2020)
Priestley duality above dimension zero: algebraic axiomatisability of the dual of compact ordered spaces.
Abstract, Slides.

Topology, Algebra, and Categories in Logic 2019. University of Nice, France. (20 June 2019)
Norm-complete Abelian ℓ-groups: equational axiomatization.
Abstract, Slides.

PhDs in Logic XI. University of Bern, Switzerland. (25 Apr 2019)
Stone-Gelfand duality for groups.
Abstract, Slides.

Boolean Algebras, Lattices, Universal Algebra, Set Theory, Topology 2018. University of Denver, CO, USA. (10 Aug 2018)
Operations that preserve integrability, and truncated Riesz spaces.
Abstract, Slides.

Contributed Talks at National Conferences

XXVII Meeting of the Italian Association for Logic and its Applications (AILA). Università della Campania "L. Vanvitelli", Caserta, Italy. (13 Sept 2022)
Duality for metrically complete Abelian ℓ-groups.
Abstract, Slides.

Talks at Local Seminars

University of Milan, Italy. (31 Oct 2024)
Quantifier-free fragments and quantifier alternation depth in doctrines.

Abstract

Step-by-step methods in algebraic logic are used to get results on the Lindenbaum-Tarski algebras with an induction on the depth of nesting of the “difficult’’ logical symbols; e.g. implications in intuitionistic propositional logic, or modalities in propositional modal logic. Step-by-step methods can be used to construct algebras satisfying a given specification in a piece-by-piece manner, understand free algebras as directed colimits of simpler partial algebras, get decidability results and normal forms.

Some of the main developments of step-by-step methods have been in intuitionistic propositional logic and propositional modal logic, where "step-by-step" refers to nesting of implications and modalities, respectively.

Our aim is to extend this approach to classical first-order logic, with "step-by-step" referring to nestings of quantifiers. \This means understanding the Lindenbaum-Tarski algebras made up of only those formulas whose quantifier alternation depth is \less than or equal to a fixed natural number, and understanding how to freely add one layer of quantifier alternation to such an algebra. I will present the first steps towards this aim.

This talk is based on a joint work with Francesca Guffanti.

Slides.

University of Manchester, UK. (9 Oct 2024)
Quantifier-free fragments and quantifier alternation depth in doctrines.

Abstract

Step-by-step methods in algebraic logic are used to get results on the Lindenbaum-Tarski algebras (= algebras of all formulas modulo logical equivalence) with an induction on the level of nesting of the “difficult’’ logical symbols; e.g. implications in intuitionistic propositional logic, or modalities in modal logic. They are used to construct algebras satisfying a given specification in a piece-by-piece manner, understand free algebras via its partial (simpler) subalgebras, get decidability results and normal forms.

The main developments of step-by-step methods have been in intuitionistic propositional logic and propositional modal logic, where the step-by-step method refers to layers of nested implications and modalities, respectively.

Our aim is to extend this approach to classical first-order logic, where the step-by-step approach refers to layers of quantifiers. This means understanding the Lindenbaum-Tarski algebras made up of only those formulas with quantifier alternation depth less than or equal to a fixed natural number, and understanding how to freely add one layer of quantifier alternation to such an algebra. We present here the first steps towards this aim.

This talk is based on a joint work with Francesca Guffanti.

Slides.

University of Padua, Italy. (2 May 2024)
Quantifier alternation depth in universal Boolean doctrines.

Abstract

In universal Boolean doctrines, by design, there is no structure that allows one to identify the quantifier depth of formulas, i.e.\ the level of nesting of quantifiers. For example, we don't have any information about what formulas are considered to be quantifier-free. In response to an invitation by M. Gehrke at the conference Category Theory 20->21, we introduce the notion of a quantifier-stratified universal Boolean doctrine. This notion requires additional structure on a universal Boolean doctrine, accounting for the quantifier alternation depth of formulas. This is motivated by the extensive usage of induction on the quantifier depth in applications related to first-order logic.

After proving that every Boolean doctrine over a small base category admits a quantifier completion, we show how to freely add the first layer of quantifier alternation depth to one such doctrine. This amounts to a doctrinal generalization of Herbrand's theorem in classical first-order logic. To achieve this version of Herbrand's theorem, we characterize, within the doctrinal setting, the classes of quantifier-free formulas whose universal closure is valid in some common model.

Slides.

University of Birmingham, UK. (23 Nov 2023)
An abstraction of the unit interval with Euclidean topology and denominators.

Abstract

Compact Hausdorff spaces are the topological abstraction of the unit interval [0,1] (in a sense that can be made precise). Let us now equip the unit interval with the "denominator map" den: [0,1] -> N that maps a rational number to its denominator and an irrational number to 0. We characterize the abstraction of [0,1] that takes into account both the topology and the denominator map.

(The reason why we were interested in this problem is that we could show that the resulting structures form a category that is categorically dual to the category of archimedean metrically complete Abelian lattice-ordered groups.)

This is a joint work with V. Marra and L. Spada. <https://arxiv.org/pdf/2210.15341.pdf>.

Slides.

Université catholique de Louvain, Louvain-la-Neuve, Belgium. (13 Nov 2023)
Soft sheaf representations in Barr-exact categories.

Abstract

It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists of 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, 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 also non-algebraic categories such as any topos.

The talk is based on: M. Abbadini, L. Reggio. Barr-exact categories and soft sheaf representations. Journal of Pure and Applied Algebra, 227(12):107413, 2023. <https://doi.org/10.1016/j.jpaa.2023.107413>.

Slides.

University of Bern, Switzerland. Bern Logic Seminar (20 Apr 2023)
Positive MV-algebras.

Abstract

MV-algebras extend the theory of Boolean algebras by replacing the two-element set of truth values {0,1} with the unit interval [0,1]. They provide the algebraic semantics of Lukasiewicz many-valued logic. Inspired by the extensive study of bounded distributive lattices, which are the negation-free subreducts of Boolean algebras, we aim to develop the theory of the negation-free subreducts of MV-algebras, called positive MV-algebras. These algebras can be thought of as the many-valued version of bounded distributive lattices. We axiomatize positive MV-algebras via finitely many quasi-equations. Moreover, generalizing Mundici's celebrated equivalence for MV-algebras [4], we obtain a categorical equivalence between positive MV-algebras and certain lattice-ordered monoids with units. We provide some results that can help to develop the theory of these algebras: in particular, we exhibit a finite quasi-equational axiomatization for the class of positive MV-algebras and a categorical equivalence with certain lattice-ordered monoids with units.

This talk is based on [1], [2], and a joint work with P. Jipsen, T. Kroupa and S. Vannucci [3].

References
[1] M. Abbadini. Equivalence à la Mundici for commutative lattice-ordered monoids. Algebra Universalis, 82:45, 2021.
[2] M. Abbadini. On the axiomatisability of the dual of compact ordered spaces. PhD thesis, University of Milan, 2021.
[3] M. Abbadini, P. Jipsen, T. Kroupa, and S. Vannucci. A finite axiomatization of positive MV-algebras. Algebra Universalis, 83:28, 2022.
[4] D. Mundici. Interpretation of AF C*-algebras in Lukasiewicz sentential calculus. J. Funct. Anal., 65(1):15–63, 1986.

Slides.

University of the Witwatersrand, Johannesburg, South Africa. (30 Mar 2023)
Unique embeddability property.

Abstract

It is well-known that every cancellative commutative monoid M can be embedded in an abelian group G (think of the embedding of ℕ into ℤ, for example), and one such embedding is essentially unique (if I embed ℕ into any abelian group, the group generated by the image will be isomorphic to ℤ). Something different happens when we remove the hypothesis of commutativity: a cancellative monoid admits non-isomorphic group envelopes. A similar phenomenon occurs in various algebraic structures close to logicians' hearts (such as bounded distributive lattices and Boolean algebras).

We give a characterisation of the algebraic structures that satisfy this unique embeddability property.

Slides.

Paris-Saclay University, Laboratoire Méthodes Formelles (LMF), Paris, France, online. (4 Jan 2023)
An abstraction of the unit interval with denominators.

Abstract

Compact Hausdorff spaces are the topological abstraction of the unit interval [0,1] (in a sense that can be made precise). Let us now equip the unit interval with the "denominator map" den: [0,1] -> N that maps a rational number to its denominator and an irrational number to 0. We characterize the abstraction of [0,1] that takes into account both the topology and the denominator map.

(We use this result to provide a representation theorem for a class of lattice-ordered groups, generalizing a result of M.H. Stone (1941).)

Slides.

University of Padua, Italy, online. (5 Dec 2022)
Dualities for compact Hausdorff spaces.

Abstract

In 1936, Stone obtained a categorical dual equivalence between Boolean algebras and Stone spaces, i.e. compact Hausdorff spaces with a basis of closed open subsets. I will review some Stone-like dualities for the larger category of compact Hausdorff spaces.

Slides.

Groupe de travail "Semantique". Institut de Recherche en Informatique Fondamentale (IRIF), Paris, France. (25 Nov 2022)
A generalization of de Vries duality to closed relations between compact Hausdorff spaces.

Abstract

De Vries (1962) obtained a categorical duality for the category KHaus of compact Hausdorff spaces and continuous functions. The objects of the dual category DeV are complete boolean algebras equipped with a proximity relation, known as de Vries algebras, and the morphisms are functions satisfying certain conditions. One drawback of DeV is that composition of morphisms is not usual function composition.

We propose an alternative approach, where morphisms between de Vries algebras are certain relations and composition of morphisms is usual relation composition. This gives a solution to the aforementioned drawback of DeV. The usage of relations as morphisms between de Vries algebras allows us to extend De Vries duality to a duality for the larger category of compact Hausdorff spaces and closed relations, solving a problem raised by G. Bezhanishvili, D. Gabelaia, J. Harding, and M. Jibladze (2019).

I will discuss the strong connections with the literature on domain theory, in particular with the notions of R-structures, information systems and abstract bases (Smyth, Scott, Abramsky, Jung, Vickers), and with other duality theoretical results (Jung, Sünderhauf, Kegelmann, Moshier).

Slides.

Lectures on Logic and its Mathematical Aspects (LLAMA seminar). Institute for Logic, Language and Computation, University of Amsterdam, online. (11 May 2022)
A generalization of the De Vries duality to compact Hausdorff spaces with closed relations.

Abstract

Stone’s representation theorem for Boolean algebras gives a bridge between algebra and topology in the form of a categorical duality. In his PhD thesis, de Vries generalized this duality to a duality between compact Hausdorff spaces and what are nowadays called de Vries algebras, which are structures that encode the set of regular open sets of a given compact Hausdorff space. One drawback of the category of de Vries algebras is that the composition of morphisms is not function composition. We propose to work with relations (rather than functions) as morphisms between de Vries algebras: this has the advantage that the composition of morphisms is usual relation composition. Moreover, this approach allows for an extension of de Vries duality to a duality for the category of compact Hausdorff spaces and closed relations between them.

Slides.

Algebra seminar. New Mexico State University, NM, USA. (14 Mar 2022)
Free extension for universal algebras.

Abstract

Given an equational class of algebras (such as groups, Boolean algebras, etc.), and a fixed sublanguage of this class (such as monoid operations, lattice operations, etc.), we can show the equivalence of two properties. The first, which is called free extension property, is more semantic: it concerns extensions of certain partial functions to homomorphisms. Whereas the second, called expressibility of equations, is concerned with terms and identities, thus being more syntactic.

Slides.

University of Caserta, Italy. (14 Feb 2022)
Positive MV-algebras.

Abstract

The Boolean terms that are order-preserving in each argument (when thought of as functions from {0,1}ⁿ to {0,1}) are precisely those expressible in the signature {∨, ∧, 0, 1}. It is known that the subreducts of Boolean algebras with respect to the signature {∨, ∧, 0, 1} are precisely the bounded distributive lattices.

We make a similar investigation replacing Boolean algebras by MV-algebras - the structures that provide the algebraic semantics of Łukasiewicz logic. The MV-terms that are order-preserving in each argument (when thought of as functions from [0,1]ⁿ to [0,1]) are precisely those expressible in the signature {⊕, ⊙, ∨, ∧, 0, 1}. In this talk we address the following question:

Is there a nice axiomatization of the subreducts of MV-algebras with respect to the signature {⊕, ⊙, ∨, ∧, 0, 1}?

We explain why an equational axiomatization is not possible, and we provide a finite quasi-equational one. To obtain this axiomatization we rely on a generalization of a theorem by Mundici connecting unital Abelian lattice-ordered groups and MV-algebras.

Nonclassical Logic Webinar. University of Denver, CO, USA, online. (5 May 2021)
Is the category of locally finite MV‑algebras equivalent to an equational class?

Abstract

Locally finite MV-algebras form a subclass of MV-algebras which is closed under homomorphic images, subalgebras, and finite products, but not under arbitrary ones. However, the category of locally finite MV-algebras with homomorphisms has arbitrary products in the classical categorical sense. Driven by these considerations, D. Mundici posed the following question:

Is the category of locally finite MV-algebras equivalent to an equational class? (D. Mundici. Advanced Lukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235, problem 3.)

We answer this question.

Our proofs rest upon the duality between locally finite MV-algebras and multisets established by R. Cignoli, E. J. Dubuc, and D. Mundici, and categorical characterizations of varieties established by J. Duskin, F. W. Lawvere, and others.

Slides.

Nonclassical Logic Seminar. University of Denver, CO, USA, online. (8 May 2020)
Equivalence à la Mundici for lattice-ordered monoids.
Abstract, Slides.

University of Salerno, Italy. (23 Jan 2020)
Equivalence à la Mundici for lattice-ordered monoids.
Abstract, Slides.

Groupe de travail `Dualité de Stone, langages formels et logique'. University of Nice, France. (3 Oct 2019)
On concrete dual adjunctions.

Applied Mathematical Logic Seminar. Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic. (6 Feb 2019)
The dual of compact ordered spaces is a variety.
Abstract, Slides.

University of Salerno, Italy. (13 Dec 2018)
Archimedean Cauchy-complete MV-algebras form a variety.
Abstract, Slides.

PhD^2 - PhD days. University of Milan, Italy. (6 July 2018)
Operations that preserve integrability: characterization and related algebraic structures.
Slides (in Italian).

Posters & Lightning talks

Lightning talk at the Autumn school ``Proof and Computation''. Fischbachau, Germany. (19 Sept 2024)
Algebras of logic, step-by-step.

Poster at the Congresso Unione Matematica Italiana 2019. University of Pavia, Italy. (2 Sept 2019)
Dualità di Stone-Gelfand per i gruppi.
Poster (in Italian).