arXiv subjects

Davide Emilio Quadrellaro

Publications and source records attributed to Davide Emilio Quadrellaro.

11 recordsLinked to original sources

arXiv2026

On the Model Theory of Open Incidence Structures: The Rank 2 Case

Taking inspiration from [1, 21, 24], we develop a general framework to deal with the model theory of open incidence structures. In this first paper we focus on the study of systems of points and lines (rank $2$). This has a number of applications, in particular we show that for any of the following classes all the non-degenerate free structures are elementarily equivalent, and their common theory is decidable, strictly stable, and with no prime model: $(k, n)$-Steiner systems (for $2 \leq k < n$); generalised $n$-gons (for $n \geq 3$); $k$-nets (for $k \geq 3$); affine planes; projective Möbius, Laguerre and Minkowski planes.

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arXiv2026

On the Model Theory of Second-Order Objects

Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that generalizes the standard notion of abstract elementary class, and show that it is an example of an accessible category. We apply our framework to show that the logic $\mathsf{FOT}$ introduced by Kontinen and Yang satisfies a version of Lindström's Theorem. Finally, we consider the problem of transferring categoricity between different cardinalities for complete theories in existential second-order logic (or independence logic) and prove both a downwards and an upwards categoricity transfer result.

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arXiv2026

A New Construction Principle

We use the framework of Abstract Elementary Classes ($\mathrm{AEC}$s) to introduce a new Construction Principle $\mathrm{CP}(\mathbf{K},\ast)$, which generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. From this we derive, in ZFC, that several uncountably categorical classes of structures are not axiomatisable in the logic $\mathfrak{L}_{\infty,ω_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{L}_{\infty,\infty}$. In particular, our methods apply to: free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank $1$ which is not $\mathbb{Q}$, free $(k,n)$-Steiner systems, and free generalised $n$-gons.

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arXiv2026

The Construction Principle and superstability of free objects in varieties of algebras

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras $\mathbf{V}$ and the question of superstability of the free objects in $\mathbf{V}$, denoted as $\mathcal{F}_\mathbf{V}$. We consider this question in the general setting of AEC-coverings of $\mathcal{F}_\mathbf{V}$, with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of $\mathcal{F}_\mathbf{V}$ are unsuperstable. Concrete applications to $R$-modules and varieties of groups are also considered.

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arXiv2026

Algebraizable Weak Logics

We extend the framework of abstract algebraic logic to weak logics, namely logical systems which are not necessarily closed under uniform substitution. We interpret weak logics by algebras expanded with an additional predicate and we introduce a loose and strict version of algebraizability for weak logics. We study this framework by investigating the connection between the algebraizability of a weak logic and the algebraizability of its schematic fragment, and we then prove a version of Blok and Pigozzi's Isomorphism Theorem in our setting. We apply this framework to logics in team semantics and show that the classical versions of inquisitive and dependence logic are strictly algebraizable, while their intuitionistic versions are only loosely so.

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arXiv2025

Borel completeness of Tits buildings with no rank 3 residues of spherical type

We prove that, for every Coxeter diagram $D$ with no rank $3$ residues of spherical type and such that $D$ has not only edges labelled by $2$, the space of countable (Tits) buildings of type $D$ is Borel complete, that is, classifying countable buildings of type $D$ up to isomorphism is as hard as classifying countable graphs up to isomorphism. In particular, for every $n\geq 3$, the space of countable generalised $n$-gons is Borel complete.

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arXiv2023

Esakia Duals of Regular Heyting Algebras

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, $\mathtt{DNA}$-logics and dependence logic.

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arXiv2023

Varieties of strictly n-generated Heyting algebras

For any $n<ω$ we construct an infinite Heyting algebra $H_n$ which is $(n+1)$-generated but that contains only finite $n$-generated subalgebras. From this we conclude that for every $n<ω$ there exists a variety of Heyting algebras which contains an infinite $(n+1)$-generated Heyting algebra, but which contains only finite $n$-generated Heyting algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [3].

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arXiv2022

Compactness in Team Semantics

We provide two proofs of the compactness theorem for extensions of first-order logic based on team semantics. First, we build upon Lück's ultraproduct construction for team semantics and prove a suitable version of Łoś' Theorem. Second, we show that by working with suitably saturated models, we can generalize the proof of Kontinen and Yang to sets of formulas with arbitrarily many variables.

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arXiv2022

On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We show they are elementary structures axiomatised by Horn sentences and we use our completeness result to state a version of algebraisability for inquisitive and dependence logics. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain Representation Theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics.

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arXiv2020

Lattices of Intermediate Theories via Ruitenburg's Theorem

For every univariate formula $χ$ we introduce a lattices of intermediate theories: the lattice of $χ$-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula $χ^2$, which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of $χ$-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.

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