Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie
{"title":"沃克取消定理的有效性","authors":"Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie","doi":"10.1002/malq.202400030","DOIUrl":null,"url":null,"abstract":"<p>Walker's cancellation theorem for abelian groups tells us that if <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> is finitely generated and <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math> are such that <span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mi>⊕</mi>\n <mi>G</mi>\n <mo>≅</mo>\n <mi>A</mi>\n <mi>⊕</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$A \\oplus G \\cong A \\oplus H$</annotation>\n </semantics></math>, then <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>≅</mo>\n <mi>H</mi>\n </mrow>\n <annotation>$G \\cong H$</annotation>\n </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>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math>, given indices for <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math>, the isomorphism between <span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mi>⊕</mi>\n <mi>G</mi>\n </mrow>\n <annotation>$A \\oplus G$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mi>⊕</mi>\n <mi>H</mi>\n </mrow>\n <annotation>$A \\oplus H$</annotation>\n </semantics></math>, and the rank of <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, is <span></span><math>\n <semantics>\n <msup>\n <mn>0</mn>\n <mo>′</mo>\n </msup>\n <annotation>$\\mathbf {0^{\\prime }}$</annotation>\n </semantics></math>. Moreover, we find that the complexity remains <span></span><math>\n <semantics>\n <msup>\n <mn>0</mn>\n <mo>′</mo>\n </msup>\n <annotation>$\\mathbf {0^{\\prime }}$</annotation>\n </semantics></math> even if the generators in the copies of <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> are specified.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202400030","citationCount":"0","resultStr":"{\"title\":\"Effectiveness of Walker's cancellation theorem\",\"authors\":\"Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie\",\"doi\":\"10.1002/malq.202400030\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Walker's cancellation theorem for abelian groups tells us that if <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> is finitely generated and <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$H$</annotation>\\n </semantics></math> are such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mi>⊕</mi>\\n <mi>G</mi>\\n <mo>≅</mo>\\n <mi>A</mi>\\n <mi>⊕</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$A \\\\oplus G \\\\cong A \\\\oplus H$</annotation>\\n </semantics></math>, then <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mo>≅</mo>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$G \\\\cong H$</annotation>\\n </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>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$H$</annotation>\\n </semantics></math>, given indices for <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$H$</annotation>\\n </semantics></math>, the isomorphism between <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mi>⊕</mi>\\n <mi>G</mi>\\n </mrow>\\n <annotation>$A \\\\oplus G$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mi>⊕</mi>\\n <mi>H</mi>\\n </mrow>\\n <annotation>$A \\\\oplus H$</annotation>\\n </semantics></math>, and the rank of <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>, is <span></span><math>\\n <semantics>\\n <msup>\\n <mn>0</mn>\\n <mo>′</mo>\\n </msup>\\n <annotation>$\\\\mathbf {0^{\\\\prime }}$</annotation>\\n </semantics></math>. Moreover, we find that the complexity remains <span></span><math>\\n <semantics>\\n <msup>\\n <mn>0</mn>\\n <mo>′</mo>\\n </msup>\\n <annotation>$\\\\mathbf {0^{\\\\prime }}$</annotation>\\n </semantics></math> even if the generators in the copies of <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> are specified.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202400030\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202400030\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202400030","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.