arXiv subjects

Marcos Mazari-Armida

Publications and source records attributed to Marcos Mazari-Armida.

23 records · Page 2Linked to original sources

arXiv2020

On superstability in the class of flat modules and perfect rings

We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left perfect. - The class of flat left $R$-modules with pure embeddings is superstable. - There exists a $λ\geq (|R| + \aleph_0)^+$ such that the class of flat left $R$-modules with pure embeddings has uniqueness of limit models of cardinality $λ$. - Every limit model in the class of flat left $R$-modules with pure embeddings is $Σ$-cotorsion. A key step in our argument is the study of limit models in the class of flat modules. We show that limit models with chains of long cofinality are cotorsion and that limit models are elementarily equivalent. We obtain a new characterization via limit models of the rings characterized in [Rot02]. We show that in these rings the equivalence between left perfect rings and superstability can be refined. We show that the results for these rings can be applied to extend [She17, 1.2] to classes of flat modules not axiomatizable in first-order logic.

math.LO↗

arXiv2020

A model theoretic solution to a problem of László Fuchs

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $λ$ such that there is a universal abelian $p$-group for purity of cardinality $λ$, i.e., an abelian $p$-group $U_λ$ of cardinality $λ$ such that every abelian $p$-group of cardinality $\leq λ$ purely embeds in $U_λ$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $λ^{\aleph_0}=λ$ or $\forall μ< λ( μ^{\aleph_0} < λ)$, then there is a universal abelian $p$-group for purity of cardinality $λ$. Moreover for $n\geq 2$, there is a universal abelian $p$-group for purity of cardinality $\aleph_n$ if and only if $2^{\aleph_0} \leq \aleph_n$. As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.

math.GR↗

arXiv2020

On universal modules with pure embeddings

We show that certain classes of modules have universal models with respect to pure embeddings. $Theorem.$ Let $R$ be a ring, $T$ a first-order theory with an infinite model extending the theory of $R$-modules and $K^T=(Mod(T), \leq_{pp})$ (where $\leq_{pp}$ stands for pure submodule). Assume $K^T$ has joint embedding and amalgamation. If $λ^{|T|}=λ$ or $\forall μ< λ( μ^{|T|} < λ)$, then $K^T$ has a universal model of cardinality $λ$. As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of $R$-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer Question 4.25 of [Maz]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

math.LO↗

arXiv2019

Algebraic description of limit models in classes of abelian groups

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality $λ$ in the class of abelian groups with the subgroup relation, then $G \cong (\oplus_λ\mathbb{Q}) \oplus \oplus_{p \text{ prime}} (\oplus_λ \mathbb{Z}(p^\infty))$. (2) If $G$ is a limit model of cardinality $λ$ in the class of torsion-free abelian groups with the pure subgroup relation, then: * If the length of the chain has uncountable cofinality, then $ G \cong (\oplus_λ \mathbb{Q} ) \oplus Π_{p\text{ prime}} \overline{(\oplus_λ \mathbb{Z}_{(p)})}$. * If the length of the chain has countable cofinality, then $G$ is not algebraically compact. We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and $(<\aleph_0)$-tame and short.

math.LO↗

arXiv2018

Universal classes near $\aleph_1$

Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, then $ψ$ has arbitrarily large models and categoricity of $ψ$ in some uncountable cardinal implies categoricity of $ψ$ in all uncountable cardinals. - If $ψ$ is categorical in $\aleph_1$, then $ψ$ is categorical in all uncountable cardinals. The theorem generalizes to the framework of $L_{ω_1, ω}$-definable tame abstract elementary classes with primes.

math.LO↗