On b Q 1 $bQ_1$ -degrees of c.e. sets

Pub Date : 2023-11-20 DOI:10.1002/malq.202300033

Roland Omanadze, Irakli Chitaia

{"title":"On \n \n \n b\n \n Q\n 1\n \n \n $bQ_1$\n -degrees of c.e. sets","authors":"Roland Omanadze, Irakli Chitaia","doi":"10.1002/malq.202300033","DOIUrl":null,"url":null,"abstract":"Using properties of simple sets we study <math>\n <semantics>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n <annotation>${bQ}_1$</annotation>\n </semantics></math>-degrees of c.e. sets. In particular, we prove: (1) If <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> are c.e. sets, <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> is a simple set and <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <msub>\n <mo>≤</mo>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n </msub>\n <mi>B</mi>\n </mrow>\n <annotation>$A\\le _{{bQ}_{1}}B$</annotation>\n </semantics></math>, then there exists a simple set <math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n <mi>C</mi>\n <msub>\n <mo>≤</mo>\n <mn>1</mn>\n </msub>\n <mi>A</mi>\n </mrow>\n <annotation>$C\\le _1 A$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>C</mi>\n <msub>\n <mo>≤</mo>\n <mn>1</mn>\n </msub>\n <mi>B</mi>\n </mrow>\n <annotation>$C\\le _1 B$</annotation>\n </semantics></math>. (2) the c.e. <math>\n <semantics>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n <annotation>${bQ}_1$</annotation>\n </semantics></math>-degrees (<math>\n <semantics>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n <annotation>${bQ}_1$</annotation>\n </semantics></math>-degrees) do not form an upper semilattice. (3) The c.e. <math>\n <semantics>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n <annotation>${bQ}_1$</annotation>\n </semantics></math>-degrees are not dense, but are upwards dense. (4) The <math>\n <semantics>\n <msub>\n <mrow>\n <mi>b</mi>\n <mi>Q</mi>\n </mrow>\n <mn>1</mn>\n </msub>\n <annotation>${bQ}_1$</annotation>\n </semantics></math>-degrees are not dense.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","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.202300033","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

Using properties of simple sets we study ${b Q}_{1}$ -degrees of c.e. sets. In particular, we prove: (1) If $A$ and $B$ are c.e. sets, $A$ is a simple set and $A \leq_{{b Q}_{1}} B$ , then there exists a simple set $C$ such that $C \leq_{1} A$ and $C \leq_{1} B$ . (2) the c.e. ${b Q}_{1}$ -degrees ( ${b Q}_{1}$ -degrees) do not form an upper semilattice. (3) The c.e. ${b Q}_{1}$ -degrees are not dense, but are upwards dense. (4) The ${b Q}_{1}$ -degrees are not dense.

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在bQ1$bQ_1$- c集合的度数上

利用简单集的性质研究了c.e.集的bQ1${bQ}_1$-度。特别地，我们证明了:(1)如果A和B是c.e.集合，A是一个简单集合，且A≤bQ1B$A\le _{{bQ}_{1}}B$，则存在一个简单集合C，使得C≤1A$C\le _1 A$且C≤1B$C\le _1 B$。(2) c.e. bQ1${bQ}_1$-degrees (bQ1${bQ}_1$-degrees)不构成上半格。(3) c.e. bQ1${bQ}_1$-度不致密，但向上致密。(4) bQ1${bQ}_1$-度不密集。

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