Layth Al‐Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie
Walker's cancellation theorem for abelian groups tells us that if is finitely generated and and are such that , then . Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between and , given indices for , , , the isomorphism between and , and the rank of , is . Moreover, we find that the complexity remains even if the generators in the copies of are specified.
{"title":"Effectiveness of Walker's cancellation theorem","authors":"Layth Al‐Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie","doi":"10.1002/malq.202400030","DOIUrl":"https://doi.org/10.1002/malq.202400030","url":null,"abstract":"Walker's cancellation theorem for abelian groups tells us that if is finitely generated and and are such that , then . Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between and , given indices for , , , the isomorphism between and , and the rank of , is . Moreover, we find that the complexity remains even if the generators in the copies of are specified.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254844","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}
Given a scale (in the sense of Shelah's pcf theory), we list various conditions ensuring that a given point is good for the scale.
给定一个标度(根据谢拉的 pcf 理论),我们列出各种条件,确保给定的点对标度有利。
{"title":"Good points for scales (and more)","authors":"Pierre Matet","doi":"10.1002/malq.202300034","DOIUrl":"https://doi.org/10.1002/malq.202300034","url":null,"abstract":"Given a scale (in the sense of Shelah's pcf theory), we list various conditions ensuring that a given point is good for the scale.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182741","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}
{"title":"Editorial correction for L. Halbeisen, R. Plati, and Saharon Shelah, “Implications of Ramsey Choice principles in ZF$mathsf {ZF}$”, https://doi.org/10.1002/malq.202300024","authors":"","doi":"10.1002/malq.202430002","DOIUrl":"https://doi.org/10.1002/malq.202430002","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182740","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}
We provide, for each natural number and each class among , , , a regular language whose associated omega‐power is complete for this class.
我们为每个自然数和ⅣⅤ类中的每个类提供一种正则表达式语言,其相关的Ω-幂对该类来说是完整的。
{"title":"Wadge degrees of Δ20$mathbf{Delta }^0_2$ omega‐powers","authors":"Olivier Finkel, Dominique Lecomte","doi":"10.1002/malq.202400024","DOIUrl":"https://doi.org/10.1002/malq.202400024","url":null,"abstract":"We provide, for each natural number and each class among , , , a regular language whose associated omega‐power is complete for this class.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182743","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}
We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group not necessarily abelian over a sufficiently saturated real closed field ; namely, that the o‐minimal universal covering group of is an open locally definable subgroup of for some ‐algebraic group (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group over , we describe as a locally definable extension of subgroups of the o‐minimal universal covering groups of commutative ‐algebraic groups (Theorem 3.4).
{"title":"Extensions of definable local homomorphisms in o‐minimal structures and semialgebraic groups","authors":"Eliana Barriga","doi":"10.1002/malq.202300028","DOIUrl":"https://doi.org/10.1002/malq.202300028","url":null,"abstract":"We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group not necessarily abelian over a sufficiently saturated real closed field ; namely, that the o‐minimal universal covering group of is an open locally definable subgroup of for some ‐algebraic group (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group over , we describe as a locally definable extension of subgroups of the o‐minimal universal covering groups of commutative ‐algebraic groups (Theorem 3.4).","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745481","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}