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Cross-Sections of Divisible Abelian $o$-Groups via Tame Pairs

Ricardo Palomino Piepenborn

TL;DR

Problem: characterize cross-sections of surjective morphisms between divisible $o$-groups. Approach: develop tameness for pairs and the standard-part map to identify internal cross-sections via cofinal, tame subgroups. Result: the images of cross-sections are exactly divisible, tame, cofinal subgroups, with the standard-part map identifying elements by their image and providing a retract. Application: translates to real closed valued fields, yielding monomial groups as tame value-group substructures and enabling a one-sorted axiomatization of RC fields with definable value group and residue field.

Abstract

It is shown that images of cross-sections of surjective morphisms $f: Γ\longrightarrow Δ$ of divisible abelian $o$-groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of $Γ$ compatible with $f$ in a suitable sense. The note concludes with an application to real closed valued fields.

Cross-Sections of Divisible Abelian $o$-Groups via Tame Pairs

TL;DR

Problem: characterize cross-sections of surjective morphisms between divisible -groups. Approach: develop tameness for pairs and the standard-part map to identify internal cross-sections via cofinal, tame subgroups. Result: the images of cross-sections are exactly divisible, tame, cofinal subgroups, with the standard-part map identifying elements by their image and providing a retract. Application: translates to real closed valued fields, yielding monomial groups as tame value-group substructures and enabling a one-sorted axiomatization of RC fields with definable value group and residue field.

Abstract

It is shown that images of cross-sections of surjective morphisms of divisible abelian -groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of compatible with in a suitable sense. The note concludes with an application to real closed valued fields.

Paper Structure

This paper contains 4 sections, 6 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on tame pairs
  3. Main Theorem
  4. An Application to Real Closed Valued Fields

Key Result

Lemma 2.6

Suppose that $A$ is tame in $B$. The standard part map $\emph{st}: \emph{c.h.}_B(A) \relbar\joinrel\twoheadrightarrow A$ is order-preserving.

Theorems & Definitions (17)

  • Definition 2.1: 1.12 in vdD/Lewenberg.T-convexity_and_tame_extensions, or pillay.definability_of_types
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 7 more