Decomposable ordered groups

TL;DR

This paper extends the theory of decomposable ordered structures to include ordered groups, proving they are supersolvable and topologically product-like, and classifies low-dimensional definable ordered groups within o-minimal fields.

Contribution

It introduces a framework for decomposable ordered groups, proves their supersolvability, and classifies low-dimensional definable ordered groups in o-minimal structures.

Findings

Any decomposable ordered group is supersolvable.

Such groups are topologically homeomorphic to products of o-minimal groups.

Complete classification of 2- and 3-dimensional definable ordered groups in o-minimal fields.

Abstract

Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.