{"title":"Limit models in strictly stable abstract elementary classes","authors":"Will Boney, Monica M. VanDieren","doi":"10.1002/malq.202200075","DOIUrl":"https://doi.org/10.1002/malq.202200075","url":null,"abstract":"<p>In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$mathcal {K}$</annotation>\u0000 </semantics></math> is an abstract elementary class satisfying\u0000\u0000 </p><p>Then for <span></span><math>\u0000 <semantics>\u0000 <mi>ϑ</mi>\u0000 <annotation>$vartheta$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>δ</mi>\u0000 <annotation>$delta$</annotation>\u0000 </semantics></math> limit ordinals <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo><</mo>\u0000 <msup>\u0000 <mi>μ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$<mu ^+$</annotation>\u0000 </semantics></math> both with cofinality <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>≥</mo>\u0000 <msubsup>\u0000 <mi>κ</mi>\u0000 <mi>μ</mi>\u0000 <mo>∗</mo>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$ge kappa ^*_mu (mathcal {K})$</annotation>\u0000 </semantics></math>, if <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$mathcal {K}$</annotation>\u0000 </semantics></math> satisfies symmetry for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>non</mi>\u0000 <mi>-</mi>\u0000 <mi>μ</mi>\u0000 <mi>-</mi>\u0000 <mi>splitting</mi>\u0000 </mrow>\u0000 <annotation>${rm non}text{-}mutext{-}{rm splitting}$</annotation>\u0000 </semantics></math> (or just <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>μ</mi>\u0000 <mo>,</mo>\u0000 <mi>δ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mu,delta)$</annotation>\u0000 </semantics></math>-symmetry), then, for any <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <annotation>$M_1$</annotation>\u0000 </semantics></math> ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"438-453"},"PeriodicalIF":0.4,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859918","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}
{"title":"Random graph coloring and the instability","authors":"Akito Tsuboi","doi":"10.1002/malq.202300043","DOIUrl":"https://doi.org/10.1002/malq.202300043","url":null,"abstract":"<p>This paper explores the coloring problem, focusing on the existence of uniformly colored substructures. The study primarily examines random graphs with edge coloring and their generic substructures. The key finding is that the absence of a monochromatic generic substructure corresponds to increased instability, meaning that the colored random graph hereditarily possesses the strict order property.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"429-437"},"PeriodicalIF":0.4,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861169","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":"Apartness relations between propositions","authors":"Zoltan A. Kocsis","doi":"10.1002/malq.202300055","DOIUrl":"https://doi.org/10.1002/malq.202300055","url":null,"abstract":"<p>We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non-trivial apartness term satisfies the weak law of excluded middle, and every Heyting algebra which contains a tight apartness term is in fact a Boolean algebra. This answers a question of Rijke regarding the correct notion of apartness for propositions, and yields a short classification of apartness terms that can occur in a Heyting algebra. We also show that Martin-Löf Type Theory is not able to construct non-trivial apartness relations between propositions.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"414-428"},"PeriodicalIF":0.4,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861168","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":"On collection schemes and Gaifman's splitting theorem","authors":"Taishi Kurahashi, Yoshiaki Minami","doi":"10.1002/malq.202400021","DOIUrl":"https://doi.org/10.1002/malq.202400021","url":null,"abstract":"<p>We study model theoretic characterizations of various collection schemes over <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>PA</mi>\u0000 <mo>−</mo>\u0000 </msup>\u0000 <annotation>$mathsf {PA}^-$</annotation>\u0000 </semantics></math> from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$n ge 0$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <mo>⊧</mo>\u0000 <msup>\u0000 <mi>PA</mi>\u0000 <mo>−</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$M models mathsf {PA}^-$</annotation>\u0000 </semantics></math>, the following are equivalent: \u0000\u0000 </p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"398-413"},"PeriodicalIF":0.4,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860882","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":"Perfectly normal nonrealcompact spaces under Martin's Maximum","authors":"Tetsuya Ishiu","doi":"10.1002/malq.202400002","DOIUrl":"https://doi.org/10.1002/malq.202400002","url":null,"abstract":"<p>We analyze the behavior of a perfectly normal nonrealcompact space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>τ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(omega _1, tau)$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <annotation>$omega _1$</annotation>\u0000 </semantics></math> such that for every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo><</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$gamma <omega _1$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mi>γ</mi>\u0000 <annotation>$gamma$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annotation>$tau$</annotation>\u0000 </semantics></math>-open and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>+</mo>\u0000 <mi>ω</mi>\u0000 </mrow>\u0000 <annotation>$gamma +omega$</annotation>\u0000 </semantics></math> is <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annotation>$tau$</annotation>\u0000 </semantics></math>-closed under Martin's Maximum. We show that there exists a club subset <span></span><math>\u0000 <semantics>\u0000 <mi>D</mi>\u0000 <annotation>$D$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <annotation>$omega _1$</annotation>\u0000 </semantics></math> such that for a stationary subset of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>δ</mi>\u0000 <mo>∈</mo>\u0000 <mo>acc</mo>\u0000 <mo>(</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$delta in operatorname{acc}(D)$</annotation>\u0000 </semantics></math>, for all <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annot","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"388-397"},"PeriodicalIF":0.4,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202400002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861588","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}
{"title":"Adding a constant and an axiom to a doctrine","authors":"Francesca Guffanti","doi":"10.1002/malq.202300053","DOIUrl":"https://doi.org/10.1002/malq.202300053","url":null,"abstract":"<p>We study the meaning of “adding a constant to a language” for any doctrine, and “adding an axiom to a theory” for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"294-332"},"PeriodicalIF":0.4,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142404689","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":"Paradoxical decompositions of free \u0000 \u0000 \u0000 F\u0000 2\u0000 \u0000 $F_2$\u0000 -sets and the Hahn-Banach axiom","authors":"Marianne Morillon","doi":"10.1002/malq.202400003","DOIUrl":"https://doi.org/10.1002/malq.202400003","url":null,"abstract":"<p>Denoting by <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$F_2$</annotation>\u0000 </semantics></math> the free group over a two-element alphabet, we show in set-theory without the axiom of choice <span></span><math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> that the existence of a (2, 2)-paradoxical decomposition of free <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$F_2$</annotation>\u0000 </semantics></math>-sets follows from the conjunction of a weakened consequence of the Hahn-Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in <span></span><math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> of a paradoxical decomposition with 4 pieces of the sphere in the 3-dimensional euclidean space follows from the same two statements restricted to the set <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math> of real numbers. Our result is linked to the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(m,n)$</annotation>\u0000 </semantics></math>-paradoxical decompositions of free <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$F_2$</annotation>\u0000 </semantics></math>-sets previously obtained by Pawlikowski (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>=</mo>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$m=n=3$</annotation>\u0000 </semantics></math>, cf. [11]) and then by Sato and Shioya (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$m=3$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n=2$</annotation>\u0000 </semantics></math>, cf. [13]) with the sole Hahn-Banach axiom.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"367-387"},"PeriodicalIF":0.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859986","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":"A note on the cardinality of definable families of sets in o-minimal structures","authors":"Pablo Andújar Guerrero","doi":"10.1002/malq.202300030","DOIUrl":"https://doi.org/10.1002/malq.202300030","url":null,"abstract":"<p>We prove that any definable family of subsets of a definable infinite set <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> in an o-minimal structure has cardinality at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>A</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$|A|$</annotation>\u0000 </semantics></math>. We derive some consequences in terms of counting definable types and existence of definable topological spaces.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 4","pages":"361-366"},"PeriodicalIF":0.4,"publicationDate":"2024-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142862029","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}
{"title":"The set of injections and the set of surjections on a set","authors":"Natthajak Kamkru, Nattapon Sonpanow","doi":"10.1002/malq.202300059","DOIUrl":"https://doi.org/10.1002/malq.202300059","url":null,"abstract":"<p>In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$mathfrak {m}$</annotation>\u0000 </semantics></math>, denoted by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>I</mi>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$I(mathfrak {m})$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>J</mi>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$J(mathfrak {m})$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>m</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathfrak {m}^mathfrak {m}$</annotation>\u0000 </semantics></math>, respectively. Among our results, we show that “<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$</annotation>\u0000 </semantics></math>”, “<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mi>J</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"275-285"},"PeriodicalIF":0.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142404725","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}
Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie
{"title":"Effectiveness of Walker's cancellation theorem","authors":"Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie","doi":"10.1002/malq.202400030","DOIUrl":"10.1002/malq.202400030","url":null,"abstract":"<p>Walker's cancellation theorem for abelian groups tells us that if <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> is finitely generated and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> are such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>G</mi>\u0000 <mo>≅</mo>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$A oplus G cong A oplus H$</annotation>\u0000 </semantics></math>, then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>≅</mo>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$G cong H$</annotation>\u0000 </semantics></math>. 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 <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math>, given indices for <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math>, the isomorphism between <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$A oplus G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$A oplus H$</annotation>\u0000 </semantics></math>, and the rank of <span></span><math>\u0000 <semantics>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"347-355"},"PeriodicalIF":0.4,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202400030","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":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
