{"title":"几乎质数模型的可判定分类谱系","authors":"N. A. Bazhenov, M. I. Marchuk","doi":"10.1007/s10469-024-09746-1","DOIUrl":null,"url":null,"abstract":"<p>We study decidable categoricity spectra for almost prime models. For any computable collection {D<sub>i</sub>}<sub>i∈ω</sub>, where D<sub>i</sub> either is a c.e. set or D<sub>i</sub> = PA, we construct a sequence of almost prime models {<span>\\({\\mathcal{M}}_{i}\\)</span>}<sub>i∈ω</sub> elementarily embedded in each other, in which case for any i there exists a finite collection of constants such that the model <span>\\({\\mathcal{M}}_{i}\\)</span> in the expansion by these constants has degree of decidable categoricity deg<sub>T</sub> (D<sub>i</sub>), if D<sub>i</sub> is a c.e. set, and has no degree of decidable categoricity if D<sub>i</sub> = PA. The result obtained extends that of S. S. Goncharov, V. Harizanov, and R. Miller [Sib. Adv. Math., 30, No. 3, 200-212 (2020)].</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 4","pages":"291 - 302"},"PeriodicalIF":0.4000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Decidable Categoricity Spectra for Almost Prime Models\",\"authors\":\"N. A. Bazhenov, M. I. Marchuk\",\"doi\":\"10.1007/s10469-024-09746-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study decidable categoricity spectra for almost prime models. For any computable collection {D<sub>i</sub>}<sub>i∈ω</sub>, where D<sub>i</sub> either is a c.e. set or D<sub>i</sub> = PA, we construct a sequence of almost prime models {<span>\\\\({\\\\mathcal{M}}_{i}\\\\)</span>}<sub>i∈ω</sub> elementarily embedded in each other, in which case for any i there exists a finite collection of constants such that the model <span>\\\\({\\\\mathcal{M}}_{i}\\\\)</span> in the expansion by these constants has degree of decidable categoricity deg<sub>T</sub> (D<sub>i</sub>), if D<sub>i</sub> is a c.e. set, and has no degree of decidable categoricity if D<sub>i</sub> = PA. The result obtained extends that of S. S. Goncharov, V. Harizanov, and R. Miller [Sib. Adv. Math., 30, No. 3, 200-212 (2020)].</p>\",\"PeriodicalId\":7422,\"journal\":{\"name\":\"Algebra and Logic\",\"volume\":\"62 4\",\"pages\":\"291 - 302\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-07-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra and Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10469-024-09746-1\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-024-09746-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
摘要
我们研究几乎质数模型的可解分类谱。对于任何可计算集合 {Di}i∈ω,其中 Di 要么是一个 c.e.集合,要么 Di = PA。在这种情况下,对于任意 i 存在一个有限的常数集合,使得由这些常数展开的模型 \({\mathcal{M}}_{i}\)具有可判定的分类度 degT (Di),如果 Di 是一个 c. e. 集合的话。e.集,而如果 Di = PA,则没有可判定分类度。所得到的结果扩展了 S. S. 冈察洛夫、V. 哈里扎诺夫和 R. 米勒[《西伯利亚高等数学》,30,第 3 期,200-212 (2020)]的结果。
Decidable Categoricity Spectra for Almost Prime Models
We study decidable categoricity spectra for almost prime models. For any computable collection {Di}i∈ω, where Di either is a c.e. set or Di = PA, we construct a sequence of almost prime models {\({\mathcal{M}}_{i}\)}i∈ω elementarily embedded in each other, in which case for any i there exists a finite collection of constants such that the model \({\mathcal{M}}_{i}\) in the expansion by these constants has degree of decidable categoricity degT (Di), if Di is a c.e. set, and has no degree of decidable categoricity if Di = PA. The result obtained extends that of S. S. Goncharov, V. Harizanov, and R. Miller [Sib. Adv. Math., 30, No. 3, 200-212 (2020)].
期刊介绍:
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.