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Order positive fields II

Margarita Korovina, Oleg Kudinov

TL;DR

The paper addresses whether the real closure of an order positive field preserves order positivity, including non-Archimedean cases. It develops a constructive framework that characterizes finite extensions and demonstrates an effective real-closure construction yielding an order positive presentation of the real closure $\overline{F}$. The main result shows that $\overline{F}$ is order positive and computably presentable, extending the scope of Ershov–Madison-style methods to a broader class of fields. This provides a practical, computable approach to obtaining order-positive real closures with implications for computable algebra and model theory.

Abstract

This paper is a part of ongoing research on order positive fields started some years ago. We prove that the real closure of an order positive field even in non-Archimedean case is also order positive.

Order positive fields II

TL;DR

The paper addresses whether the real closure of an order positive field preserves order positivity, including non-Archimedean cases. It develops a constructive framework that characterizes finite extensions and demonstrates an effective real-closure construction yielding an order positive presentation of the real closure . The main result shows that is order positive and computably presentable, extending the scope of Ershov–Madison-style methods to a broader class of fields. This provides a practical, computable approach to obtaining order-positive real closures with implications for computable algebra and model theory.

Abstract

This paper is a part of ongoing research on order positive fields started some years ago. We prove that the real closure of an order positive field even in non-Archimedean case is also order positive.
Paper Structure (4 sections, 6 theorems, 14 equations)

This paper contains 4 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Basic notions and definitions
  3. Topological consistency
  4. The real closure of an order positive field

Key Result

Proposition 1

(korkud_al_1) If $(F,\alpha)$ is an effective algebra and $\alpha^{-1}(<)$ is computably enumerable then one can effectively construct a numbering $\beta$ induced by $\alpha$ such that $(F,\beta)$ satisfies the conditions (1--3).

Theorems & Definitions (9)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Theorem 1