Related papers: On the abstract elementary class of acts with embe…
We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such…
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…
We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming…
This paper is a contribution to "neo-stability" type of result for abstract elementary classes. Under certain set theoretic assumptions, we propose a definition and a characterization of NIP in AECs. The class of AECs with NIP properly…
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.…
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…
The main purpose of this paper is investigating classes of acts that are injective relative to all embeddings with indecomposable domains or codomains. We give some homological classifications of monoids in light of such kinds of…
Our main result (Theorem 1) suggests a possible dividing line ($\mu$-superstable $+$ $\mu$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of…
We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting…
In this paper we prove: Theorem 1. Let $\mathcal{K}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\lambda>\mu\geq LS(\mathcal{K})$ and $\theta$ is a limit ordinal $<\lambda^+$. If…
We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition know as tameness (currently…
For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\lambda^+$-minimal types and…
We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of…
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…
The paper is devoted to the investigation of uniform notion for acts over semigroups perceived as an overclass of subdirectly irreducible acts. We establish conditions to fill the gap between these classes of acts. Besides we prove that…
This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of…
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…
We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and…
Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion…
We prove that for any homotopy type $X$, there is an abstract elementary class $\mathcal{C}$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$. We provide a few explicit…