Small extensions of abelian ordered groups

TL;DR

This paper constructs a comprehensive classification of small extensions of totally ordered abelian groups using Hahn's embedding theorem, with applications to valuations on polynomial rings.

Contribution

It introduces the small-extensions closure Gamma_{ ext{sme}} and demonstrates its completeness and significance in valuation theory.

Findings

Gamma_{ ext{sme}} is complete and classifies small extensions.

The construction aids in understanding valuations on polynomial rings.

The approach links ordered group theory with valuation classification.

Abstract

Let $\Gamma$ be a totally ordered group. We use Hahn's embedding theorem to construct a totally ordered set $\Gamma\subset \Gamma_{\operatorname{sme}}$ which classifies small extensions of $\Gamma$. This small-extensions closure $\Gamma_{\operatorname{sme}}$ is complete and plays a crucial role in the description of equivalence classes of valuations on the polynomial ring $K[x]$ over a field $K$.