Remarks on theories of free algebras and modules

A remark about amalgamation of union of theories

We provide conditions under which the union of two first-order theories has the amalgamation property.

Remarks on generic stability in independent theories

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as $ϕ$-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate in…

余挠理论与分裂器

Cotorsion Theories and Splitters

设R是有理数的子环。我们想研究自分裂R-模G，即Ext_R（G，G）=0成立，并沿用Schultz的方法称这类模为分裂器.自由模和无挠余挠模是分裂器的经典例子。还有其他人吗通过对Schultz的一个公开问题的讨论，我们证明了存在更多的分裂子，事实上，我们能够用自由R-模结构来定义它们的自同态R-代数。作为一个副产品，我们能够回答一个问题的Salce表明，所有合理的cotortion理论有足够的内射和足够的projectives。

Remarks on Barr's theorem: Proofs in geometric theories

A theorem, usually attributed to Barr, yields that (A) geometric implications deduced in classical L_{\inftyω} logic from geometric theories also have intuitionistic proofs. Barr's theorem is of a topos-theoretic nature and its proof is non-constructive. In the literature one also finds mysterious comments about the capacity of this theorem to remove the axiom of choice from derivations. This article investigates the proof-theoretic side of Barr's theorem and also aims to shed some light on the axiom of choice part. More concretely, a constructive proof of the Hauptsatz for L_{\inftyω} is give…

模理论的Grothendieck环

Grothendieck Rings of Theories of Modules

模型论的Grothendieck环的一阶结构，定义的Krajiclouk和Scanlon，捕捉一些组合性质的可定义子集的有限权力的结构。本文计算了右$R$ -模$M$的Grothendieck环，$K_0(M_\mathcal R)$，其中$\mathcal R$是任何单位环.作为推论，我们证明了Prest的一个猜想：当$M$非零时，$K_0(M)$是非平凡的.主要的证明使用了单纯复形的同调论中的各种技术。

Two Remarks on First-Order Theories of Baumslag-Solitar Groups

In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.