{"title":"Th (N，·)$ \\operatorname{Th}(\\mathbb {N}，\\cdot)$","authors":"Atticus Stonestrom","doi":"10.1002/malq.202100049","DOIUrl":null,"url":null,"abstract":"<p>‘Skolem arithmetic’ is the complete theory <i>T</i> of the multiplicative monoid <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>N</mi>\n <mo>,</mo>\n <mo>·</mo>\n <mo>)</mo>\n </mrow>\n <annotation>$(\\mathbb {N},\\cdot )$</annotation>\n </semantics></math>. We give a full characterization of the <math>\n <semantics>\n <mi>⌀</mi>\n <annotation>$\\varnothing$</annotation>\n </semantics></math>-definable stably embedded sets of <i>T</i>, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that <i>T</i> has weak elimination of imaginaries but not elimination of finite imaginaries.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100049","citationCount":"1","resultStr":"{\"title\":\"Some model theory of \\n \\n \\n Th\\n (\\n N\\n ,\\n ·\\n )\\n \\n $\\\\operatorname{Th}(\\\\mathbb {N},\\\\cdot )$\",\"authors\":\"Atticus Stonestrom\",\"doi\":\"10.1002/malq.202100049\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>‘Skolem arithmetic’ is the complete theory <i>T</i> of the multiplicative monoid <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>N</mi>\\n <mo>,</mo>\\n <mo>·</mo>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(\\\\mathbb {N},\\\\cdot )$</annotation>\\n </semantics></math>. We give a full characterization of the <math>\\n <semantics>\\n <mi>⌀</mi>\\n <annotation>$\\\\varnothing$</annotation>\\n </semantics></math>-definable stably embedded sets of <i>T</i>, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that <i>T</i> has weak elimination of imaginaries but not elimination of finite imaginaries.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-04-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100049\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202100049\",\"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.202100049","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Some model theory of
Th
(
N
,
·
)
$\operatorname{Th}(\mathbb {N},\cdot )$
‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid . We give a full characterization of the -definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.
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