{"title":"Computable scott sentences for quasi–Hopfian finitely presented structures","authors":"Gianluca Paolini","doi":"10.1007/s00153-022-00833-7","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that every quasi-Hopfian finitely presented structure <i>A</i> has a <i>d</i>-<span>\\(\\Sigma _2\\)</span> Scott sentence, and that if in addition <i>A</i> is computable and <i>Aut</i>(<i>A</i>) satisfies a natural computable condition, then <i>A</i> has a computable <i>d</i>-<span>\\(\\Sigma _2\\)</span> 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 <i>d</i>-<span>\\(\\Sigma _2\\)</span> Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable <i>d</i>-<span>\\(\\Sigma _2\\)</span> 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 <i>d</i>-<span>\\(\\Sigma _2\\)</span> Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).\n\n\n</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-022-00833-7.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-022-00833-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
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).
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.