{"title":"局部零最小结构和维数的预几何","authors":"Masato Fujita","doi":"10.1002/malq.202200069","DOIUrl":null,"url":null,"abstract":"<p>We define a discrete closure operator for definably complete locally o-minimal structures <math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math>. The pair of the underlying set of <math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it <math>\n <semantics>\n <mo>discl</mo>\n <annotation>$\\operatorname{discl}$</annotation>\n </semantics></math>-dimension. A definable set <i>X</i> is of dimension equal to the <math>\n <semantics>\n <mo>discl</mo>\n <annotation>$\\operatorname{discl}$</annotation>\n </semantics></math>-dimension of <i>X</i>. The structure <math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure <math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> also coincides with its dimension.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Pregeometry over locally o-minimal structures and dimension\",\"authors\":\"Masato Fujita\",\"doi\":\"10.1002/malq.202200069\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We define a discrete closure operator for definably complete locally o-minimal structures <math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$\\\\mathcal {M}$</annotation>\\n </semantics></math>. The pair of the underlying set of <math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$\\\\mathcal {M}$</annotation>\\n </semantics></math> and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it <math>\\n <semantics>\\n <mo>discl</mo>\\n <annotation>$\\\\operatorname{discl}$</annotation>\\n </semantics></math>-dimension. A definable set <i>X</i> is of dimension equal to the <math>\\n <semantics>\\n <mo>discl</mo>\\n <annotation>$\\\\operatorname{discl}$</annotation>\\n </semantics></math>-dimension of <i>X</i>. The structure <math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$\\\\mathcal {M}$</annotation>\\n </semantics></math> is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure <math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$\\\\mathcal {M}$</annotation>\\n </semantics></math> also coincides with its dimension.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200069\",\"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.202200069","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Pregeometry over locally o-minimal structures and dimension
We define a discrete closure operator for definably complete locally o-minimal structures . The pair of the underlying set of and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it -dimension. A definable set X is of dimension equal to the -dimension of X. The structure is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure also coincides with its dimension.