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Mutual algebraicity and cellularity 互代数性和细胞性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-02-18 DOI: 10.1007/s00153-021-00804-4
Samuel Braunfeld, Michael C. Laskowski
{"title":"Mutual algebraicity and cellularity","authors":"Samuel Braunfeld,&nbsp;Michael C. Laskowski","doi":"10.1007/s00153-021-00804-4","DOIUrl":"10.1007/s00153-021-00804-4","url":null,"abstract":"<div><p>We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure <i>M</i> is cellular if and only if <i>M</i> is <span>(omega )</span>-categorical and mutually algebraic. Second, if a countable structure <i>M</i> in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50035655","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: 5
Equivalence of generics 泛型的等价性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-24 DOI: 10.1007/s00153-021-00813-3
Iian B. Smythe
{"title":"Equivalence of generics","authors":"Iian B. Smythe","doi":"10.1007/s00153-021-00813-3","DOIUrl":"10.1007/s00153-021-00813-3","url":null,"abstract":"<div><p>Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, focusing on Cohen and random forcing. We prove, among other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, and hence is amenable, while the latter is neither amenable nor treeable.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45280963","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
Hindman’s theorem for sums along the full binary tree, (Sigma ^0_2)-induction and the Pigeonhole principle for trees 沿全二叉树的和的Hindman定理，(Sigma ^0_2) -归纳法和树的鸽子洞原理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-21 DOI: 10.1007/s00153-021-00814-2
Lorenzo Carlucci, Daniele Tavernelli
{"title":"Hindman’s theorem for sums along the full binary tree, (Sigma ^0_2)-induction and the Pigeonhole principle for trees","authors":"Lorenzo Carlucci,&nbsp;Daniele Tavernelli","doi":"10.1007/s00153-021-00814-2","DOIUrl":"10.1007/s00153-021-00814-2","url":null,"abstract":"<div><p>We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to <span>(Sigma ^0_2)</span>-induction over <span>(mathsf {RCA}_0)</span>. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees <span>({mathsf {T},}{mathsf {T}}^1)</span> with an extra condition on the solution tree.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-021-00814-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50040683","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
Hindman’s theorem for sums along the full binary tree, $$Sigma ^0_2$$ Σ 2 0 -induction and the Pigeonhole principle for trees 沿全二叉树的和的Hindman定理，$$Sigma ^0_2$$ Σ 20 -归纳和树的鸽子洞原理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-21 DOI: 10.1007/s00153-021-00814-2
L. Carlucci, Daniele Tavernelli
{"title":"Hindman’s theorem for sums along the full binary tree, \u0000 \u0000 \u0000 \u0000 $$Sigma ^0_2$$\u0000 \u0000 \u0000 Σ\u0000 2\u0000 0\u0000 \u0000 \u0000 -induction and the Pigeonhole principle for trees","authors":"L. Carlucci, Daniele Tavernelli","doi":"10.1007/s00153-021-00814-2","DOIUrl":"https://doi.org/10.1007/s00153-021-00814-2","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42724081","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 isomorphism problem for some classes of computable algebraic structures 关于一类可计算代数结构的同构问题
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-20 DOI: 10.1007/s00153-021-00811-5
Valentina S. Harizanov, Steffen Lempp, Charles F. D. McCoy, Andrei S. Morozov, Reed Solomon
{"title":"On the isomorphism problem for some classes of computable algebraic structures","authors":"Valentina S. Harizanov,&nbsp;Steffen Lempp,&nbsp;Charles F. D. McCoy,&nbsp;Andrei S. Morozov,&nbsp;Reed Solomon","doi":"10.1007/s00153-021-00811-5","DOIUrl":"10.1007/s00153-021-00811-5","url":null,"abstract":"<div><p>We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is <span>(Sigma _{1}^{1})</span>-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47122445","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
Reverse mathematics and semisimple rings 逆向数学和半单环
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-17 DOI: 10.1007/s00153-021-00812-4
Huishan Wu
{"title":"Reverse mathematics and semisimple rings","authors":"Huishan Wu","doi":"10.1007/s00153-021-00812-4","DOIUrl":"10.1007/s00153-021-00812-4","url":null,"abstract":"<div><p>This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that <span>(mathrm ACA_{0})</span> is equivalent to the statement that any left module over a left semisimple ring is semisimple over <span>(mathrm RCA_{0})</span>. We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: (1) <span>(mathrm ACA_{0})</span> is equivalent to the statement that any left module over a left semisimple ring is projective over <span>(mathrm RCA_{0})</span>; (2) <span>(mathrm ACA_{0})</span> is equivalent to the statement that any left module over a left semisimple ring is injective over <span>(mathrm RCA_{0})</span>; (3) <span>(mathrm RCA_{0})</span> proves the statement that if every cyclic left <i>R</i>-module is projective, then <i>R</i> is a left semisimple ring; (4) <span>(mathrm ACA_{0})</span> proves the statement that if every cyclic left <i>R</i>-module is injective, then <i>R</i> is a left semisimple ring.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50034131","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
Reverse mathematics and semisimple rings 逆向数学和半单环
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-17 DOI: 10.1007/s00153-021-00812-4
Huishan Wu
{"title":"Reverse mathematics and semisimple rings","authors":"Huishan Wu","doi":"10.1007/s00153-021-00812-4","DOIUrl":"https://doi.org/10.1007/s00153-021-00812-4","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52098990","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
Iterated multiplication in ( VTC ^0) 迭代乘法 ( VTC ^0)
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-04 DOI: 10.1007/s00153-021-00810-6
Emil Jeřábek
{"title":"Iterated multiplication in ( VTC ^0)","authors":"Emil Jeřábek","doi":"10.1007/s00153-021-00810-6","DOIUrl":"10.1007/s00153-021-00810-6","url":null,"abstract":"<div><p>We show that <span>( VTC ^0)</span>, the basic theory of bounded arithmetic corresponding to the complexity class <span>(mathrm {TC}^0)</span>, proves the <span>( IMUL )</span> axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the <span>(mathrm {TC}^0)</span> iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, <span>( VTC ^0)</span> can also prove the integer division axiom, and (by our previous results) the <span>( RSUV )</span>-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories <span>(Delta ^b_1text{- } CR )</span> and <span>(C^0_2)</span>. As a side result, we also prove that there is a well-behaved <span>(Delta _0)</span> definition of modular powering in <span>(IDelta _0+ WPHP (Delta _0))</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50007938","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: 3
Small (mathfrak {u}(kappa )) at singular (kappa ) with compactness at (kappa ^{++}) 小的(mathfrak {u}(kappa ))在奇异的(kappa )具紧致在 (kappa ^{++})
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2022-01-01 DOI: 10.1007/s00153-021-00776-5
Radek Honzik, Šárka Stejskalová
{"title":"Small (mathfrak {u}(kappa )) at singular (kappa ) with compactness at (kappa ^{++})","authors":"Radek Honzik,&nbsp;Šárka Stejskalová","doi":"10.1007/s00153-021-00776-5","DOIUrl":"10.1007/s00153-021-00776-5","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00153-021-00776-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50050591","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 note on cut-elimination for classical propositional logic 经典命题逻辑的切消注释
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2021-11-26 DOI: 10.1007/s00153-021-00800-8
G. Pulcini
{"title":"A note on cut-elimination for classical propositional logic","authors":"G. Pulcini","doi":"10.1007/s00153-021-00800-8","DOIUrl":"https://doi.org/10.1007/s00153-021-00800-8","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52098503","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
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