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Tapani Hyttinen

Publications and source records attributed to Tapani Hyttinen.

At least 19 recordsLinked to original sources
arXiv2026

The lattice of abstract elementary classes of modules

Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.

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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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arXiv2025

Generalized Descriptive Set Theory and Classification Theory

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

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arXiv2025

On Borel subsets of generalized Baire spaces

We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $κ^{<κ}=κ$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to obtain a meaningful theory. This modification has no effect if $κ^{<κ}=κ$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $κ$ in the space $κ^κ$ are $κ$-Borel in our generalized sense. It turns out that this question depends, as is the case when $κ^{<κ}=κ$, on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.

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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

On highly equivalent non-isomorphic countable models of arithmetic and set theory

It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $α$. We also prove the same result for models of ZFC.

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arXiv2022

On ultraproducts, the spectral theorem and rigged Hilbert spaces

We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators $K$ by calculating them in the finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle$.

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arXiv2022

An AEC framework for fields with commuting automorphisms

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA,the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms,the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms.We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models,that the class is homogeneous,and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).

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arXiv2020

First-Order Model Theory of Free Projective Planes

We prove that the theory of open projective planes is complete and strictly stable, and infer from this that Marshall Hall's free projective planes $(π^n : 4 \leq n \leq ω)$ are all elementary equivalent and that their common theory is strictly stable and decidable, being in fact the theory of open projective planes. We further characterize the elementary substructure relation in the class of open projective planes, and show in particular that $(π^n : 4 \leq n \leq ω)$ is an elementary chain. We then prove that the theory of open projective planes does not have a prime model, that it has elimination of quantifiers down to Boolean combinations of existential formulas, and that it is not model complete. Finally, we characterize the forking independence relation in models of the theory and prove that the $π^n$'s ($4 \leq n \leq ω)$ are strongly type-homogeneous.

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arXiv2019

On $Σ_1^1$-completeness of quasi-orders on $κ^κ$

We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $Σ_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($κ>ω$). This improvement to known results in $L$ has many new consequences concerning the $Σ_1^1$-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in literature. Additionally the theorem is applied to prove a dichotomy in $L$: If the isomorphism of a countable first-order theory (not necessarily complete) is not $Δ_1^1$, then it is $Σ_1^1$-complete. We also study the case $V\ne L$ and prove $Σ_1^1$-completeness results for weakly ineffable and weakly compact $κ$.

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arXiv2018

Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translate into a machinery to build concrete examples of $\mathrm{AECs}$ satisfying given model-theoretic properties. We exhibit the power of our method constructing three concrete examples of finitary classes. We show that the first and third class are non-homogeneous, and that the last two are tame, uncountably categorical and axiomatizable by a single $L_{ω_{1}, ω}$-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is $κ$-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

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arXiv2018

On Eigenvectors, Approximations and the Feynman Propagator

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $\langle x_{0}| K^{t}| x_{1}\rangle $ of the eigenvector $| x_{0}\rangle $ of the position operator with eigenvalue $x_{0}$ and $K^{t}(| x_{1}\rangle )$, where $K^{t}$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and without heavy case-by-case scaling, it gives the wrong value. We look at this phenomenon, and then complement this by showing how to instead calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.

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arXiv2018

Categoricity and Universal Classes

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=λ$ categorical in regular $κ>λ^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>λ}$ for which there is $\mathcal{B} \in \mathcal{K}_{\geκ}$ such that $\mathcal{A}\subseteq\mathcal{B}$. We prove that $\mathcal{K}^*$ is categorical in every $ξ>λ^+$, $\mathcal{K}_{\ge\beth_{(2^{λ^+})^+}} \subseteq \mathcal{K}^{*}$, and the models of $\mathcal{K}^*_{>λ^+}$ are essentially vector spaces (or trivial i.e. disintegrated).

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arXiv2017

Reduction of Database Independence to Dividing in Atomless Boolean Algebras

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the former case of independence can be seen as the discrete version of the latter.

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arXiv2017

Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $(\mathbf{K}^3, \preccurlyeq)$, is $ω$-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $(\mathbf{K}^3, \preccurlyeq)$ the Smoothness Axiom fails, and so $(\mathbf{K}^3, \preccurlyeq)$ is not an $\mathrm{AEC}$.

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