I study distributive lattices, Priestley duality, profinite semigroups, model theory, and more recently also automata and programming languages.
If you want to get a first impression, below is a selection of slides I made for presentations about my research. You can also take a look at my full list of publications below.
Slides

Preserving joins at primes: a connection between lattices, domains and automata.
About a newly identified property of implications, or bilinear maps, on distributive lattices, that turns up in both domain theory and profinite monoids.
LX group meeting, LaBRI, Bordeaux and DuaLL workshop, Nice, 2022.

Priestley duality for MV algebras and beyond.
About algebras for Lukaciewicz’ multivalued logic, focusing on duality, sheaf representations and elementary characterizations.
BLAST conference, New Mexico, online, 2021.

What is an existentially closed Heyting algebra and what does it have to do with automata?
This is an open question; the slides explain why I ask it.
Dresden QuantLA seminar, 2020 and India Webinar in Logic, online, 2021.

Model theory and proaperiodic monoids.
About a new approach to studying profinite monoids, by seeing their elements as pseudofinite models of a firstorder theory.
Seminar on Semigroups, Automata and Languages, Porto, online, 2020.

Pointlike sets for varieties determined by groups.
A new result on pointlike sets and separation of regular languages by FO+MODdefinable languages.
Diamant Symposium, Veenendaal, 2018.

Machines, Models, Monoids, and Modal Logic.
A sequence of tutorial lectures on connections between these fields.
Tbilisi Symposium on Logic, Languages, and Computation, 2017.

A history of talks and conferences, with some older slides, is
here.
Publications
2022

Textbook, preprint, 2022
This book is a course in StonePriestley duality theory, with applications to logic and theoretical computer science. Our target audience are graduate students and researchers in mathematics and computer science. Our aim is to get in a fairly full palette of duality tools as directly and quickly as possible, then to illustrate and further elaborate these tools within the setting of three emblematic applications: semantics of propositional logics, domain theory in logical form, and the theory of profinite monoids for the study of regular languages and automata. This preprint version contains the first part of the book, a graduate level ‘crash course’ in duality theory as it is practiced now, and a chapter on applications to domain theory.

Foundations of Software Science and Computation Structures (FoSSaCS) 2022
We show that the existence of a firstorder formula separating two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on ωwords. For this, we develop the algebraic concept of monoid (resp. ωsemigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. ωsemigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by firstorder indistinguishability. We also show the computability of FOpointlike sets, and the decidability of the covering problem for firstorder logic on countable ordinal words.
2021

Relational and Algebraic Methods in Computer Science (RAMiCS) 2021
Graded modalities have been proposed in recent work on programming languages as a general framework for refining type systems with intensional properties. In particular, continuous endomaps of the discrete time scale, or time warps, can be used to quantify the growth of information in the course of program execution. Time warps form a complete residuated lattice, with the residuals playing an important role in potential programming applications. In this paper, we study the algebraic structure of time warps, and prove that their equational theory is decidable, a necessary condition for their use in realworld compilers. We also describe how our universalalgebraic proof technique lends itself to a constraintbased implementation, establishing a new link between universal algebra and verification technology.

Forum Mathematicum 2021
We provide a new perspective on extended Priestley duality for large class of distributive lattices equipped with binary double quasioperators. Under this approach, nonlattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as firstorder conditions on dual spaces. In particular, we specialize our general results to the variety of MValgebras, obtaining a duality for these in which the equations axiomatizing MValgebras are dualized as firstorder conditions.
2019

Advances in Mathematics 2019
For a variety of finite groups H, let Hbar denote the variety of finite semigroups all of whose subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to Hbar. Our characterization is effective whenever H has a decidable membership problem. In particular, the separation problem for Hbarlanguages is decidable for any decidable variety of finite groups H. This generalizes Henckell’s theorem on decidability of aperiodic pointlikes.

Israel Journal of Mathematics 2019
We apply Stone duality and model theory to study the structure theory of free proaperiodic monoids. Stone duality implies that elements of the free proaperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ωterms that avoids using McCammond’s normal forms, as well as new proofs and extensions of other structural results concerning free proaperiodic monoids.

Canadian Mathematical Bulletin 2019
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the twosided KrohnRhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup T into a twosided semidirect product whose components are built from two subsemigroups T₁, T₂, which together generate T, and the subsemigroup generated by their setwise product T₁T₂. In this sense we decompose T by merging the subsemigroups T₁ and T₂. More generally, our technique merges semigroup homomorphisms from free semigroups.

Research note, 2019
An unpublished note (work in progress) on extending our work on saturated models to prime models.
2018

Research note, 2018
A logic satisfies the interpolation property provided that whenever a formula Δ is a consequence of another formula Γ, then this is witnessed by a formula Θ which only refers to the language common to Γ and Δ. That is, the relational (and functional) symbols occurring in Θ occur in both Γ and Δ, Γ has Θ as a consequence, and Θ has Δ as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate Gödel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate Gödel logic. While this has no impact on settling the question for predicate Gödel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.

Topology and its Applications 2018
We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, selfcontained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras.

Journal of Pure and Applied Algebra 2018
It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a onetoone correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributivelatticeordered algebras this allows us to dualize such sheaf representations.
2017

Journal of Symbolic Logic 2017
Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., firstorder models based on the natural numbers with order, successor, and finitely many unary predicate symbols.
Monadic second order logic over infinite words (S1S) can alternatively be described as a firstorder logic interpreted in 𝒫(ω), the power set Boolean algebra of the natural numbers, equipped with modal operators for ’initial’, ’next’ and ’future’ states. We prove that the firstorder theory of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic (LTL) without until. The proof makes crucial use of two classical, nontrivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1Sformulas and Büchi automata.

Annals of Pure and Applied Logic 2017
Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown, following related results of Ghilardi and Zawadowski, that a combination of these properties provides a sufficient condition for the firstorder theory of the variety to admit a model completion.

Symposium on Theoretical Aspects of Computer Science (STACS) 2017
We apply Stone duality and model theory to study the structure theory of free proaperiodic monoids. Stone duality implies that elements of the free proaperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for omegaterms that avoids using McCammond’s normal forms, as well as new proofs and extensions of other structural results concerning free proaperiodic monoids.
2016

Logic in Computer Science (LICS) 2016
The main focus of this paper is on bisimulationinvariant MSO,
and more particularly on giving a novel modeltheoretic approach
to it. In model theory, a model companion of a theory is a first
order description of the class of models in which all potentially
solvable systems of equations and nonequations have solutions.
We show that bisimulationinvariant MSO on trees gives the model
companion for a new temporal logic, “fair CTL”, an enrichment
of CTL with local fairness constraints. To achieve this, we give a
completeness proof for the logic fair CTL which combines tableaux
and Stone duality, and a fair CTL encoding of the automata for the
modal μcalculus. Moreover, we also show that MSO on binary
trees is the model companion of binary deterministic fair CTL.
2015

Tbilisi Symposium on Language, Logic and Computation (TbiLLC) 2015
2014

Sam van Gool
Advances in Modal Logic 2014
We give a construction of finitely generated free algebras for GödelLöb provability
logic, GL. On the semantic side, this construction yields a notion of canonical graded
model for GL and a syntactic definition of those normal forms which are consistent
with GL. Our two main techniques are incremental constructions of free algebras
and finite duality for partial modal algebras. In order to apply these techniques to
GL, we use a rulebased formulation of the logic GL by Avron (which we simplify
slightly), and the corresponding semantic characterization that was recently obtained
by Bezhanishvili and Ghilardi.

Journal of Algebra 2014
We study representations of MValgebras — equivalently, unital latticeordered abelian groups — through the lens of StonePriestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (StonePriestley) dual spaces of MValgebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MVspectrum, the other over the maximal MVspectrum. These decompositions yield sheaf representations of MValgebras, using a new and purely dualitytheoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MValgebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MValgebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the StonePriestley duals of MValgebras. In addition, using our analysis of these decompositions, we prove that MValgebras with isomorphic underlying lattices have homeomorphic maximal MVspectra. This result is an MValgebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0,1]valued functions on the spaces are isomorphic.

On sheaves and duality.
Sam van Gool
PhD thesis, Radboud University Nijmegen 2014
2013

Order 2013
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of ‘admissibility’ to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasiuniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.

Topology and its Applications 2013
We prove that the category of lefthanded strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a noncommutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.
From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and socalled override. Our duality shows that there is a canonical choice for this embedding.
Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.

Journal of Logic and Computation 2013
In this paper we introduce a new setting, based on partial algebras, for studying con
structions of finitely generated free algebras. We give sufficient conditions under which
the finitely generated free algebras for a variety V may be described as the colimit of a
chain of finite partial algebras obtained by repeated application of a functor. In particular,
our method encompasses the construction of finitely generated free algebras for varieties of
algebras for a functor as in [2], Heyting algebras as in [1] and S4 algebras as in [8].
2012

Sam van Gool
Topology and its Applications 2012
We construct a canonical extension for strong proximity lattices in order to give an algebraic, pointfree description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct π and σextensions.
2010

Logic, Language and Meaning 2010
This paper examines how intonation affects the interpretation of disjunctive questions. The semantic effect of a question is taken to be threefold. First, it raises an issue. In the tradition of inquisitive semantics, we model this by assuming that a question proposes several possible updates of the common ground (several possibilities for short) and invites other participants to help establish at least one of these updates. But apart from raising an issue, a question may also highlight and/or suggest certain possibilities, and intonation determines to a large extent which possibilities are highlighted/suggested.
Collaborations
Science is hard to do alone.
Adrien Guatto (Paris), Hugo Férée (Paris), Thomas Colcombet (Paris), Wesley Fussner (Nice), Mikolaj Bojanczyk (Warsaw), Denis Kuperberg (Lyon), Rosalie Iemhoff (Utrecht, postdoc advisor), Yde Venema (Amsterdam), Benjamin Steinberg (New York, postdoc advisor), Luca Reggio (Oxford), Silvio Ghilardi (Milan, postdoc advisor), Costas Tsinakis (Nashville), George Metcalfe (Bern, postdoc advisor), Daniela Petrisan (Paris), PaulAndré Melliès (Paris, postdoc advisor), Michael Pinsker (Vienna), Andrej Bauer (Ljubljana), Karin Cvetko Vah (Ljubljana), Ganna Kudryavtseva (Ljubljana), Vincenzo Marra (Milan), Dick de Jongh (Amsterdam), Nick Bezhanishvili (Amsterdam), Dion Coumans (Nijmegen), Mai Gehrke (Nice, PhD advisor), Alessandra Palmigiano (Amsterdam, MSc advisor)
People I worked with when they were students (I learned a lot from all of them):
 Anatole Leterrier (Master’s, IRIF, ongoing), on completeness theorems for temporal logics on trees.

Jérémie Marques (PhD, U. Nice, ongoing),
on hyperdoctrines and StonePriestley duality.

Vincent Moreau (Master’s, IRIF, 2021; PhD, IRIF, ongoing), I am coadvising Vincent with PaulAndré Melliès on canonical extensions, profinite monoids, and higher order automata.

Rémi Morvan (Master’s, IRIF, 2021), I coadvised Rémi with Thomas Colcombet on a project on separation for infinite words; Rémi is now a PhD student at LaBRI, Bordeaux.

Axel Osmond (PhD, IRIF, 2021), I worked together with Axel and Mai Gehrke on a project on sheaves and duality.

Mehdi Zaïdi (PhD, U. Nice, ongoing), I worked together with Mehdi and Mai Gehrke on formal languages and duality.
 Thomas Randriamahazaka (Master’s, LMFI, ENS PSL, 2020), I supervised Thomas’ Master’s project on distributive Birkhoff systems, a kind of lattices without absorption; Thomas is now a PhD student at St. Andrews.
 Corto Mascle (Master’s, MPRI, ENS ParisSaclay, 2020), I coadvised Corto with Thomas Colcombet on relating pointlike sets and logic; Corto is now a PhD student at LaBRI, Bordeaux.
 Clément Chivet (Bachelor’s, ENS PSL, 2020), on the finite model property for intuitionistic logics.
 Chase Ford (Master’s, M. Logic, ILLC, U. Amsterdam, 2019), I coadvised Chase with Yde Venema on automata for logics on trees; Chase is now a PhD student at FAU ErlangenNürnberg.

Luca Reggio (PhD, IRIF, 2018), during his PhD I worked with Luca on a topological proof of uniform interpolation for intuitionistic logic; Luca was a Marie SklodowskaCurie Postdoctoral Fellow at Oxford University and is now a senior research fellow at University College London.