{"title":"Order Positive Fields. I","authors":"M. V. Korovina, O. V. Kudinov","doi":"10.1007/s10469-024-09738-1","DOIUrl":null,"url":null,"abstract":"<p>The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, we show that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 3","pages":"203 - 214"},"PeriodicalIF":0.4000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-024-09738-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
The notion of a computable structure based on numberings with decidable equality is well established with a number of prominent results. Nevertheless, applied to strictly ordered fields, it fails to capture some natural properties and constructions for which decidability of equality is not assumed. For example, the field of primitive recursive real numbers is not computable, and there exists a computable real closed field with noncomputable maximal Archimedean subfields. We introduce the notion of an order positive field which aims to overcome these limitations. A general criterion is presented which decides when an Archimedean field is order positive. Using this criterion, we show that the field of primitive recursive real numbers is order positive and that the Archimedean parts of order positive real closed fields are order positive. We also state a program for further research.
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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