Computable Scott sentences and the weak Whitehead problem for finitely presented groups

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2024-03-20 DOI:10.1016/j.apal.2024.103441

Gianluca Paolini

引用次数: 0

Abstract

We prove that if A is a computable Hopfian finitely presented structure, then A has a computable d- $Σ_{2}$ Scott sentence if and only if the weak Whitehead problem for A is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable d- $Σ_{2}$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its $\exists^{+}$ -types, a question which arose in a different context.

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有限呈现群的可计算斯科特句子和弱怀特海问题

我们证明，如果是可计算的霍普菲有限呈现结构，那么只有当弱怀特海问题可解时，才有可计算的斯科特句子。我们以此推断出，每个双曲群和任何多环无限群都有一个可计算的斯科特句子，从而涵盖了有限呈现群的两大类。我们的证明还意味着，每个弱霍普菲有限呈现群都是由其-类型强定义的，这是在不同背景下出现的一个问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.