TY - JOUR

T1 - Order Positive Fields. I

AU - Korovina, M. V.

AU - Kudinov, O. V.

N1 - O. V. Kudinov is supported by RSCF grant No. 23-11-00170 (https://rscf.ru/project/23-11-00170) and by the Program of Fundamental Research RAS, project FWNF-2022-0011. Публикация для корректировки.

PY - 2023/7

Y1 - 2023/7

N2 - 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.

AB - 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.

KW - computable real numbers

KW - positive structures

KW - strictly ordered fields

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85191061720&origin=inward&txGid=353097177c29889f59f03de64cad22b1

UR - https://www.mendeley.com/catalogue/8b9a7ef0-649d-3aea-8412-1de5efd1cc10/

U2 - 10.1007/s10469-024-09738-1

DO - 10.1007/s10469-024-09738-1

M3 - Article

VL - 62

SP - 203

EP - 214

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 3

ER -