Computable scott sentences for quasi–Hopfian finitely presented structures

IF 0.3 4区数学 Q1 Arts and Humanities

Archive for Mathematical Logic Pub Date : 2022-06-25 DOI:10.1007/s00153-022-00833-7

Gianluca Paolini

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引用次数: 1

Abstract

We prove that every quasi-Hopfian finitely presented structure A has a d-\(\Sigma _2\) Scott sentence, and that if in addition A is computable and Aut(A) satisfies a natural computable condition, then A has a computable d-\(\Sigma _2\) Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable d-\(\Sigma _2\) Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable d-\(\Sigma _2\) Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank 4 has a computable d-\(\Sigma _2\) Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).

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拟Hopfian有限表示结构的可计算scott语句

我们证明了每一个准hopfian有限呈现结构A都有一个d- \(\Sigma _2\) Scott句，并且如果A是可计算的并且Aut(A)满足一个自然可计算的条件，那么A就有一个可计算的d- \(\Sigma _2\) Scott句。这统一了关于有限呈现结构的Scott句的几个已知结果，并用于证明其他以前未考虑的感兴趣的代数结构具有可计算的d- \(\Sigma _2\) Scott句。特别地，我们证明了每个有限秩的直角Coxeter群都有一个可计算的d- \(\Sigma _2\) Scott句，以及任何有限秩的强刚性Coxeter群。最后，我们证明了秩4的自由投影平面有一个可计算的d- \(\Sigma _2\) Scott句子，从而展示了一个使用准hopfianity假设的自然例子(因为这个结构不是Hopfian)。

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

Archive for Mathematical Logic MATHEMATICS-LOGIC

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.