Cuts and small extensions of abelian ordered groups

TL;DR

This paper classifies cuts in ordered abelian groups, analyzes their properties, and constructs real vector spaces to understand extensions of valuations from valued fields to rational function fields.

Contribution

It provides a classification of cuts in ordered abelian groups, computes coinitiality and cofinality for divisible cases, and constructs ordered real vector spaces to study valuation extensions.

Findings

Classified cuts in ordered abelian groups.

Computed coinitiality and cofinality for divisible groups.

Constructed ordered real vector spaces for valuation extension analysis.

Abstract

We classify cuts in (totally) ordered abelian groups $\g$ and compute the coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with small extensions of $\g$ in a natural way, which leads to an explicit construction of a totally ordered real vector space containing realizations of all cuts. This construction is applied to the problem of classifying all extensions of the valuation from a given valued field $K$ to the rational function field $K(x)$.