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Random structures and automorphisms with a single orbit 具有单轨道的随机结构和自同构
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-14 DOI: 10.1007/s00153-024-00944-3
Hirotaka Kikyo, Akito Tsuboi
{"title":"Random structures and automorphisms with a single orbit","authors":"Hirotaka Kikyo,&nbsp;Akito Tsuboi","doi":"10.1007/s00153-024-00944-3","DOIUrl":"10.1007/s00153-024-00944-3","url":null,"abstract":"<div><p>We investigate the class of <i>m</i>-hypergraphs whose substructures with <i>l</i> elements have more than <i>s</i> <i>m</i>-element subsets that do not form a hyperedge. The class will have the free amalgamation property if <i>s</i> is small, but it does not if <i>s</i> is large. We find the boundary of <i>s</i>. Suppose the class has the free amalgamation property. In the case <span>(m ge 3)</span>, we demonstrate that the random structure for the class has continuum-many automorphisms with a single orbit. The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"423 - 433"},"PeriodicalIF":0.3,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00944-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bounded distributive lattices with strict implication and weak difference 具有严格蕴涵和弱差分的有界分配格
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-13 DOI: 10.1007/s00153-024-00945-2
Sergio Celani, Agustín Nagy, Botero William Zuluaga
{"title":"Bounded distributive lattices with strict implication and weak difference","authors":"Sergio Celani,&nbsp;Agustín Nagy,&nbsp;Botero William Zuluaga","doi":"10.1007/s00153-024-00945-2","DOIUrl":"10.1007/s00153-024-00945-2","url":null,"abstract":"<div><p>In this paper we introduce the class of weak Heyting–Brouwer algebras (WHB-algebras, for short). We extend the well known duality between distributive lattices and Priestley spaces, in order to exhibit a relational Priestley-like duality for WHB-algebras. Finally, as an application of the duality, we build the tense extension of a WHB-algebra and we employ it as a tool for proving structural properties of the variety such as the finite model property, the amalgamation property, the congruence extension property and the Maehara interpolation property.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"387 - 422"},"PeriodicalIF":0.3,"publicationDate":"2024-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A characterization of strongly computable finite factorization domains 强可计算有限因式分解域的表征
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-12 DOI: 10.1007/s00153-024-00941-6
Geraldo Soto-Rosa, Victor Ocasio-González
{"title":"A characterization of strongly computable finite factorization domains","authors":"Geraldo Soto-Rosa,&nbsp;Victor Ocasio-González","doi":"10.1007/s00153-024-00941-6","DOIUrl":"10.1007/s00153-024-00941-6","url":null,"abstract":"<div><p>In recent research, the prime and irreducible elements of strong finite factorization domains were studied. It was shown that strongly computable strong finite factorization domains (SCSFFD) have necessarily computable irreducible elements and a computable division algorithm. However, the question of how to best classify this class of structures is left unanswered. This work provides a classification for SCSFFDs by showing the existence of a computable norm where norm-form equations can be solved computably. This classification provides the intuition to extend further the notion of strong computability to finite factorization domains in general.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"333 - 349"},"PeriodicalIF":0.3,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the non-measurability of (omega )-categorical Hrushovski constructions 论(omega ) -范畴赫鲁晓夫斯基结构的不可测性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-09 DOI: 10.1007/s00153-024-00943-4
Paolo Marimon
{"title":"On the non-measurability of (omega )-categorical Hrushovski constructions","authors":"Paolo Marimon","doi":"10.1007/s00153-024-00943-4","DOIUrl":"10.1007/s00153-024-00943-4","url":null,"abstract":"<div><p>We study <span>(omega )</span>-categorical <i>MS</i>-measurable structures. Our main result is that a certain class of <span>(omega )</span>-categorical Hrushovski constructions, supersimple of finite <i>SU</i>-rank is not <i>MS</i>-measurable. These results complement the work of Evans on a conjecture of Macpherson and Elwes. In constrast to Evans’ work, our structures may satisfy independent <i>n</i>-amalgamation for all <i>n</i>. We also prove some general results in the context of <span>(omega )</span>-categorical <i>MS</i>-measurable structures. Firstly, in these structures, the dimension in the <i>MS</i>-dimension-measure can be chosen to be <i>SU</i>-rank. Secondly, non-forking independence implies a form of probabilistic independence in the measure. The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"351 - 386"},"PeriodicalIF":0.3,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00943-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Different covering numbers of compact tree ideals 紧凑树理想的不同覆盖数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-16 DOI: 10.1007/s00153-024-00933-6
Jelle Mathis Kuiper, Otmar Spinas
{"title":"Different covering numbers of compact tree ideals","authors":"Jelle Mathis Kuiper,&nbsp;Otmar Spinas","doi":"10.1007/s00153-024-00933-6","DOIUrl":"10.1007/s00153-024-00933-6","url":null,"abstract":"<div><p>We investigate the covering numbers of some ideals on <span>({^{omega }}{2}{})</span> associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, <span>({mathfrak {C}_{2}})</span>, remain small in the Sacks model.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"259 - 278"},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00933-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On categoricity of scattered linear orders of constructive ranks 论构造等级的分散线性阶的分类性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-16 DOI: 10.1007/s00153-024-00934-5
Andrey Frolov, Maxim Zubkov
{"title":"On categoricity of scattered linear orders of constructive ranks","authors":"Andrey Frolov,&nbsp;Maxim Zubkov","doi":"10.1007/s00153-024-00934-5","DOIUrl":"10.1007/s00153-024-00934-5","url":null,"abstract":"<div><p>In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals the successor of a limit ordinal is still open.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"279 - 297"},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The provably total functions of basic arithmetic and its extensions 基本算术及其扩展的可证实总函数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-14 DOI: 10.1007/s00153-024-00939-0
Mohammad Ardeshir, Erfan Khaniki, Mohsen Shahriari
{"title":"The provably total functions of basic arithmetic and its extensions","authors":"Mohammad Ardeshir,&nbsp;Erfan Khaniki,&nbsp;Mohsen Shahriari","doi":"10.1007/s00153-024-00939-0","DOIUrl":"10.1007/s00153-024-00939-0","url":null,"abstract":"<div><p>We study Basic Arithmetic, <span>(textsf{BA})</span> introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). <span>(textsf{BA})</span> is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of <span>(textsf{BA})</span> is a <i>proper</i> sub-class of the primitive recursive functions. Three extensions of <span>(textsf{BA})</span>, called <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span> and <span>(textsf{EBA})</span> are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of <span>(textsf{BA})</span> are <i>exactly</i> the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in <span>(textsf{BA})</span>, <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span>, but holds in <span>(textsf{EBA})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"205 - 257"},"PeriodicalIF":0.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Undecidability of indecomposable polynomial rings 不可分解多项式环的不可判定性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-12 DOI: 10.1007/s00153-024-00936-3
Marco Barone, Nicolás Caro-Montoya, Eudes Naziazeno
{"title":"Undecidability of indecomposable polynomial rings","authors":"Marco Barone,&nbsp;Nicolás Caro-Montoya,&nbsp;Eudes Naziazeno","doi":"10.1007/s00153-024-00936-3","DOIUrl":"10.1007/s00153-024-00936-3","url":null,"abstract":"<div><p>By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to <span>Raphael Robinson</span>, which deals with a restricted class of polynomial integral domains.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"185 - 203"},"PeriodicalIF":0.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141940768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Punctually presented structures II: comparing presentations 按时展示的结构 II：展示比较
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-08 DOI: 10.1007/s00153-024-00940-7
Marina Dorzhieva, Rodney Downey, Ellen Hammatt, Alexander G. Melnikov, Keng Meng Ng
{"title":"Punctually presented structures II: comparing presentations","authors":"Marina Dorzhieva,&nbsp;Rodney Downey,&nbsp;Ellen Hammatt,&nbsp;Alexander G. Melnikov,&nbsp;Keng Meng Ng","doi":"10.1007/s00153-024-00940-7","DOIUrl":"10.1007/s00153-024-00940-7","url":null,"abstract":"<div><p>We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order (<span>(mathbb {Z},&lt;)</span>) are dense. Finally we characterise the Boolean algebras which have a punctually 1-decidable presentation that is computably isomorphic to a 1-decidable presentation.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"159 - 184"},"PeriodicalIF":0.3,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00940-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141928836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On some (Sigma ^{B}_{0})-formulae generalizing counting principles over (V^{0}) 关于在 $$V^{0}$ 上概括计数原理的一些 $$Sigma ^{B}_{0}$ 公式
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-07-22 DOI: 10.1007/s00153-024-00938-1
Eitetsu Ken
{"title":"On some (Sigma ^{B}_{0})-formulae generalizing counting principles over (V^{0})","authors":"Eitetsu Ken","doi":"10.1007/s00153-024-00938-1","DOIUrl":"10.1007/s00153-024-00938-1","url":null,"abstract":"<div><p>We formalize various counting principles and compare their strengths over <span>(V^{0})</span>. In particular, we conjecture the following mutual independence between:</p><ul>\u0000 <li>\u0000 <p>a uniform version of modular counting principles and the pigeonhole principle for injections,</p>\u0000 </li>\u0000 <li>\u0000 <p>a version of the oddtown theorem and modular counting principles of modulus <i>p</i>, where <i>p</i> is any natural number which is not a power of 2,</p>\u0000 </li>\u0000 <li>\u0000 <p>and a version of Fisher’s inequality and modular counting principles.</p>\u0000 </li>\u0000 </ul><p> Then, we give sufficient conditions to prove them. We give a variation of the notion of <i>PHP</i>-tree and <i>k</i>-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have <i>o</i>(<i>n</i>)-evaluations. As for the remaining two, we utilize well-known notions of <i>p</i>-tree and <i>k</i>-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"117 - 158"},"PeriodicalIF":0.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00938-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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