arXiv subjects

Marcos Mazari-Armida

Publications and source records attributed to Marcos Mazari-Armida.

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.

math.LO↗

arXiv2026

On the abstract elementary class of acts with pure embeddings

We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid $S$ being LO (for every $s,t \in S$, we have that $s \in St$ or $t \in Ss$) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when $S$ is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when $S$ is LO.

math.LO↗

arXiv2026

Deconstructible classes of modules and stability

We show that every deconstructible class of modules with all embeddings, all pure embedding and all RD-embeddings is stable. The argument is presented in the context of abstract classes of modules without amalgamation and the key idea is to construct a stable-like independence relation. In particular, the following classes of modules with all embeddings, all pure embedding and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each $n \geq 0$, the classes of all modules of projective and flat dimension $\leq n$ over any ring, and the class of all modules of injective dimension $\leq n$ over any right noetherian ring.

math.LO↗

arXiv2026

Examples of non-tame abstract elementary classes of abelian groups

We construct an abstract elementary class $K_1$ of torsion-free abelian groups such that $K_1$ is not $(<\aleph_0)$-tame but is $\aleph_0$-tame. This answers a question of [BoVa17]. Furthermore, for every regular uncountable cardinal $μ$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^μ)$ of torsion-free abelian groups such that $K_2(2^μ)$ is not $(<μ)$-tame. $K_1$ and $K_2(2^μ)$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language.

math.LO↗

arXiv2026

Cofibrant generation of pure monomorphisms in presheaf categories

We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant generation of pure monomorphisms between sets with an $S$-action in terms of $S$: this happens if and only if for all $a, b \in S$ there is $c \in S$ such that $a = cb$ or $ca = b$. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.

math.CT↗

arXiv2026

An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example

We construct a class $\hat{K}$ of torsion-free abelian groups such that $\hat{\mathbf{K}}=(\hat{K}, \leq_p)$ is an abstract elementary class with $\operatorname{LS}(\hat{\mathbf{K}})=\aleph_0$ such that: $(\cdot)$ $\hat{\mathbf{K}}$ is not stable; $(\cdot)$ $\hat{\mathbf{K}}$ has the joint embedding property and no maximal models, but does not have the amalgamation property; $(\cdot)$ $\hat{\mathbf{K}}$ is $(<\aleph_0)$-tame. The class we construct is a variation of [PaSh, Section 4] which isolates the core mechanism of the Paolini-Shelah construction.

math.LO↗

arXiv2026

On the abstract elementary class of acts with embeddings

We study the class of acts with embeddings as an abstract elementary class. We show that the class is always stable and show that superstability in the class is characterized algebraically via weakly noetherian monoids. The study of these model-theoretic notions and limit models lead us to introduce parametized weakly noetherian monoids and find a characterization of them via parametrized injective acts. Furthermore, we obtain a characterization of weakly noetherian monoids via absolutely pure acts extending a classical result of ring theory. The paper is aimed at algebraists and model theorists so an effort was made to provide the background for both.

math.LO↗

arXiv2025

On the spectrum of limit models

We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic. $\textbf{Theorem.}$ Let $\mathbf{K}$ be a $\aleph_0$-tame abstract elementary class stable in $λ\geq \operatorname{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $λ$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\geq κ)$-local character in a minimal regular $κ< λ^+$. Suppose $δ_1, δ_2 < λ^+$ with $\operatorname{cf}(δ_1) < \operatorname{cf}(δ_2)$. Then for any $N_1, N_2, M \in \mathbf{K}_λ$ where $N_l$ is a $(λ, δ_l)$-limit model over $M$ for $l = 1, 2$, \[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(δ_1) \geq κ\] Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.

math.LO↗

arXiv2025

On limit models and parametrized noetherian rings

We study limit models in the abstract elementary class of modules with embeddings as algebraic objects. We characterize parametrized noetherian rings using the degree of injectivity of certain limit models. We show that the number of limit models and how close a ring is from being noetherian are inversely proportional. $\textbf{Theorem.}$ Let $n \geq 0$ The following are equivalent. 1. $R$ is left $(<\aleph_{n } )$-noetherian but not left $(< \aleph_{n -1 })$-noetherian. 2.The abstract elementary class of modules with embeddings has exactly $n +1$ non-isomorphic $λ$-limit models for every $λ\geq (\operatorname{card}(R) + \aleph_0)^+$ such that the class is stable in $λ$. We further show that there are rings such that the abstract elementary class of modules with embeddings has exactly $κ$ non-isomorphic $λ$-limit models for every infinite cardinal $κ$.

math.RA↗

arXiv2024

On Stability and Existence of Models in Abstract Elementary Classes

For an abstract elementary class $\mathbf{K}$ and a cardinal $λ\geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $λ^+$-minimal types and continuity of splitting in $λ$, that stability in $λ$ is equivalent to the existence of a model in $λ^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $λ$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models.

math.LO↗

arXiv2024

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, $\mathcal{M}$-submodules and $\mathcal{M}$-epimorphic images where $\mathcal{M}$ is either the class of embeddings, $RD$-embeddings or pure embeddings. We show that the $\mathcal{M}$-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, $RD$-injective modules, pure injective modules, flat cotorsion modules and $\mathfrak{s}$-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah's non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

math.RA↗

arXiv2024

Building models in small cardinals in local abstract elementary classes

There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{stability} is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{Theorem.}$ Suppose $λ<2^{\aleph_0}$. Let $\mathbf{K}$ be an abstract elementary class with $λ\geq LS(\mathbf{K})$. Assume $\mathbf{K}$ has amalgamation in $λ$, no maximal model in $λ$, and is stable in $λ$. If $\mathbf{K}$ is $(<λ^+, λ)$-local, then $\mathbf{K}$ has a model of cardinality $λ^{++}$. The set theoretic assumption that $λ<2^{\aleph_0}$ and model theoretic assumption of stability in $λ$ can be weakened to the model theoretic assumptions that $|\mathbf{S}^{na}(M)|< 2^{\aleph_0}$ for every $M \in \mathbf{K}_λ$ and stability for $λ$-algebraic types in $λ$. This is a significant improvement of Theorem 0.1., as the result holds on some unstable abstract elementary classes.

math.LO↗

arXiv2023

A note on torsion modules with pure embeddings

We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the $Σ$-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative $Σ$-pure-injective modules strictly extends the classical charactetization of [GrJe76] and [Zim, 3.6]. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a pure submodule. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable, i.e., stable not superstable.

math.LO↗

arXiv2023

A countable universal torsion abelian group for purity

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below $\aleph_ω$. We introduce $\aleph_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.

math.GR↗

arXiv2022

Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $λ$-categorical in $all$ $λ> |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring. The class of absolutely pure $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R$ is a local artinian ring. We show that in the above results it is enough to assume $λ$-categoricity in $some$ large cardinal $λ$. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings. We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.

math.RA↗

arXiv2021

Some stable non-elementary classes of modules

Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of modules and $\leq_p$ is the pure submodule relation. In this paper we give some instances where this is true: $\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K, \leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed under finite direct sums, then: - If $K$ is closed under pure-injective envelopes, then $(K, \leq_p)$ is $λ$-stable for every $λ\geq LS(K)$ such that $λ^{|R| + \aleph_0}= λ$. - If $K$ is closed under pure submodules and pure epimorphic images, then $(K, \leq_p)$ is $λ$-stable for every $λ$ such that $λ^{|R| + \aleph_0}= λ$. - Assume $R$ is Von Neumann regular. If $K$ is closed under submodules and has arbitrarily large models, then $(K, \leq_p)$ is $λ$-stable for every $λ$ such that $λ^{|R| + \aleph_0}= λ$. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy. Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of injective torsion modules.

math.LO↗

arXiv2021

Simple-like independence relations in abstract elementary classes

We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. $Theorem$: Let $K$ be an AEC with a monster model. - If $K$ has a simple independence relation, then $K$ does not have the 2-tree property. - If $K$ has a simple independence relation with $(<\aleph_0)$-witness property, then $K$ does not have the tree property. The proof of both facts is done by finding cardinal bounds to classes of small Galois-types over a fixed model that are inconsistent for large subsets. We think this finer way of counting types is an interesting notion in itself. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

math.LO↗

arXiv2020

Superstability, noetherian rings and pure-semisimple rings

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We characterize noetherian rings via superstability of the class of left modules with embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left noetherian. - The class of left $R$-modules with embeddings is superstable. - For every $λ\geq |R| + \aleph_0$, there is $χ\geq λ$ such that the class of left $R$-modules with embeddings has uniqueness of limit models of cardinality $χ$. - Every limit model in the class of left $R$-modules with embeddings is $Σ$-injective. We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left pure-semisimple. - The class of left $R$-modules with pure embeddings is superstable. - There exists $λ\geq (|R| + \aleph_0)^+$ such that the class of left $R$-modules with pure embeddings has uniqueness of limit models of cardinality $λ$. - Every limit model in the class of left $R$-modules with pure embeddings is $Σ$-pure-injective. We think that both equivalences provide evidence that that the notion of superstability could shed light in the understanding of algebraic concepts. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

math.LO↗