Ordered abelian groups that do not have elimination of imaginaries

TL;DR

This paper explores the elimination of imaginaries in ordered abelian groups, showing that some Hahn products lack this property, while certain finite lexicographic products have definable Skolem functions when constants are added.

Contribution

It demonstrates non-elimination of imaginaries in specific Hahn products and establishes the existence of definable Skolem functions in finite lexicographic products with added constants.

Findings

Hahn products of ordered abelian groups do not eliminate imaginaries in pure language.

Finite lexicographic products $ Z^n$ and $ Z^n imes Q$ have definable Skolem functions with added constants.

Abstract

We investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. We show that certain Hahn products of ordered abelian groups do not eliminate imaginaries in the pure language of ordered groups. Moreover, we prove that, adding finitely many constants to the language of ordered abelian groups, the theories of the finite lexicographic products $\mathbb{Z}^n$ and $\mathbb{Z}^n \times \mathbb{Q}$ have definable Skolem functions.