Pub Date : 2024-07-30DOI: 10.1007/s10469-024-09746-1
N. A. Bazhenov, M. I. Marchuk
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)].
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。
{"title":"Decidable Categoricity Spectra for Almost Prime Models","authors":"N. A. Bazhenov, M. I. Marchuk","doi":"10.1007/s10469-024-09746-1","DOIUrl":"10.1007/s10469-024-09746-1","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":0.4,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-27DOI: 10.1007/s10469-024-09752-3
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-024-09752-3","DOIUrl":"10.1007/s10469-024-09752-3","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 4","pages":"376 - 377"},"PeriodicalIF":0.4,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142414270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-27DOI: 10.1007/s10469-024-09751-4
N. T. Kogabaev
{"title":"Complexity of the Problem of ∀-Representation for Sentences","authors":"N. T. Kogabaev","doi":"10.1007/s10469-024-09751-4","DOIUrl":"10.1007/s10469-024-09751-4","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 4","pages":"372 - 375"},"PeriodicalIF":0.4,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141797562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-09DOI: 10.1007/s10469-024-09739-0
H. R. O. Ribeiro, H. L. Mariano
A multiring is a ring-like structure where the sum is multivalued and a hyperring is a multiring with a strong distributive property. With every multiring we associate a structural presheaf, and when that presheaf is a sheaf, we say that the multiring is geometric. A characterization of geometric von Neumann hyperrings is presented. And we build a von Neumann regular hull for multirings which is used in applications to algebraic theory of quadratic forms. Namely, we describe the functor Q, introduced by M. Marshall in [J. Pure Appl. Alg., 205, No. 2, 452-468 (2006)], as a left adjoint functor for the natural inclusion of the category of real reduced multirings (similar to real semigroups) into the category of preordered multirings and explore some of its properties. Next, we employ sheaf-theoretic methods to characterize real reduced hyperrings as certain geometric von Neumann regular real hyperrings and construct the functor V , a geometric von Neumann regular hull for a multiring. Finally, we look at some interesting logical and algebraic interactions between the functors Q and V that are useful for describing hyperrings in the image of the functor Q and that will allow us to explore the theory of quadratic forms for (formally) real semigroups.
多环是一种类似环的结构，其中的和是多值的，而超环是一种具有强分配性质的多环。每一个多接线都与一个结构预设相关联，当这个预设是一个剪子时，我们就说这个多接线是几何的。我们提出了几何冯-诺依曼超环的特征。我们还为多环建立了一个冯-诺依曼正则壳，并将其应用于二次型代数理论。也就是说，我们描述了马歇尔（M. Marshall）在[J. Pure Appl. Alg.接下来，我们用剪子理论的方法把实还原超环表征为某些几何冯-诺依曼正则实超环，并构造多环的几何冯-诺依曼正则壳的函子 V。最后，我们将探讨函子 Q 和 V 之间一些有趣的逻辑和代数相互作用，这些相互作用对于描述函子 Q 的像中的超环非常有用，并将使我们能够探索（形式上的）实半群的二次型理论。
{"title":"Von Neumann Regular Hyperrings and Applications to Real Reduced Multirings","authors":"H. R. O. Ribeiro, H. L. Mariano","doi":"10.1007/s10469-024-09739-0","DOIUrl":"10.1007/s10469-024-09739-0","url":null,"abstract":"<p>A multiring is a ring-like structure where the sum is multivalued and a hyperring is a multiring with a strong distributive property. With every multiring we associate a structural presheaf, and when that presheaf is a sheaf, we say that the multiring is geometric. A characterization of geometric von Neumann hyperrings is presented. And we build a von Neumann regular hull for multirings which is used in applications to algebraic theory of quadratic forms. Namely, we describe the functor Q, introduced by M. Marshall in [J. Pure Appl. Alg., <b>205</b>, No. 2, 452-468 (2006)], as a left adjoint functor for the natural inclusion of the category of real reduced multirings (similar to real semigroups) into the category of preordered multirings and explore some of its properties. Next, we employ sheaf-theoretic methods to characterize real reduced hyperrings as certain geometric von Neumann regular real hyperrings and construct the functor V , a geometric von Neumann regular hull for a multiring. Finally, we look at some interesting logical and algebraic interactions between the functors Q and V that are useful for describing hyperrings in the image of the functor Q and that will allow us to explore the theory of quadratic forms for (formally) real semigroups.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"215 - 256"},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1007/s10469-024-09738-1
M. V. Korovina, O. V. Kudinov
The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, we show that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.
{"title":"Order Positive Fields. I","authors":"M. V. Korovina, O. V. Kudinov","doi":"10.1007/s10469-024-09738-1","DOIUrl":"10.1007/s10469-024-09738-1","url":null,"abstract":"<p>The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, we show that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"203 - 214"},"PeriodicalIF":0.4,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140675957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-19DOI: 10.1007/s10469-024-09742-5
L. L. Maksimova, V. F. Yun
{"title":"Pretabularity and Craig’s Interpolation Property","authors":"L. L. Maksimova, V. F. Yun","doi":"10.1007/s10469-024-09742-5","DOIUrl":"10.1007/s10469-024-09742-5","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"277 - 282"},"PeriodicalIF":0.4,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140682811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-19DOI: 10.1007/s10469-024-09745-2
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-024-09745-2","DOIUrl":"10.1007/s10469-024-09745-2","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"289 - 290"},"PeriodicalIF":0.4,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-19DOI: 10.1007/s10469-024-09743-4
V. V. Rybakov
{"title":"Multi-Agent Temporal Logics, Information, Unification, and Projectivity","authors":"V. V. Rybakov","doi":"10.1007/s10469-024-09743-4","DOIUrl":"10.1007/s10469-024-09743-4","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"283 - 288"},"PeriodicalIF":0.4,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140683156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-19DOI: 10.1007/s10469-024-09744-3
J. Tang, N. Yang, A. S. Mamontov
We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., 54, No. 1, 114-118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present work can be used in studying groups with a given set of element orders, also called a spectrum.
我们证明了 A. S. Mamontov [Sib. Math. J., 54, No. 1, 114-118 (2013)]论文中关于周期为 12 的群中 Baer-Suzuki 定理有效性的定理 4。本工作的结果可用于研究具有给定元素阶集（也称为谱）的群。
{"title":"The Baer–Suzuki Theorem for Groups of 3-Exponent 1","authors":"J. Tang, N. Yang, A. S. Mamontov","doi":"10.1007/s10469-024-09744-3","DOIUrl":"10.1007/s10469-024-09744-3","url":null,"abstract":"<p>We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., <b>54</b>, No. 1, 114-118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present work can be used in studying groups with a given set of element orders, also called a spectrum.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"266 - 271"},"PeriodicalIF":0.4,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-19DOI: 10.1007/s10469-024-09741-6
R. A. Kozlov
{"title":"Faithful Representations of Finite Type for Conformal Lie Algebras","authors":"R. A. Kozlov","doi":"10.1007/s10469-024-09741-6","DOIUrl":"10.1007/s10469-024-09741-6","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"272 - 276"},"PeriodicalIF":0.4,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140685134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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