F. Rakymzhankyzy, N. A. Bazhenov, A. A. Issakhov, B. S. Kalmurzayev
{"title":"Minimal Generalized Computable Numberings and Families of Positive Preorders","authors":"F. Rakymzhankyzy, N. A. Bazhenov, A. A. Issakhov, B. S. Kalmurzayev","doi":"10.1007/s10469-022-09688-6","DOIUrl":null,"url":null,"abstract":"<div><div><p>We study A-computable numberings for various natural classes of sets. For an arbitrary oracle A≥<sub>T</sub><b>0</b><sup>′</sup>, an example of an A-computable family S is constructed in which each A-computable numbering of S has a minimal cover, and at the same time, S does not satisfy the sufficient conditions for the existence of minimal covers specified in [Sib. Math. J., <b>43</b>, No. 4, 616-622 (2002)]. It is proved that the family of all positive linear preorders has an A-computable numbering iff A<sup>′</sup>≥<sub>T</sub><b>0</b><sup>\"</sup>. We obtain a series of results on minimal A-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.</p></div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-022-09688-6","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
We study A-computable numberings for various natural classes of sets. For an arbitrary oracle A≥T0′, an example of an A-computable family S is constructed in which each A-computable numbering of S has a minimal cover, and at the same time, S does not satisfy the sufficient conditions for the existence of minimal covers specified in [Sib. Math. J., 43, No. 4, 616-622 (2002)]. It is proved that the family of all positive linear preorders has an A-computable numbering iff A′≥T0". We obtain a series of results on minimal A-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.
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