Elimination of imaginaries in Ordered Abelian groups with bounded regular rank

TL;DR

This paper investigates elimination of imaginaries in ordered abelian groups with bounded regular rank, achieving weak elimination with added quotient sorts and full elimination in the dp-minimal case with additional constants.

Contribution

It introduces a method to eliminate imaginaries in ordered abelian groups with bounded regular rank by adding specific quotient sorts, and fully eliminates imaginaries in the dp-minimal case with constants.

Findings

Weak elimination of imaginaries with quotient sorts for bounded regular rank groups.

Complete elimination of imaginaries in the dp-minimal case with added constants.

Provides a framework for understanding imaginaries in ordered abelian groups.

Abstract

In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/ \Delta+ l\Gamma$ where $\Delta$ is a definable convex subgroup and $l \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the \emph{quotient sorts}. For the dp-minimal case we obtain a complete elimination of imaginaries, if we also add constants to distinguish the elements of the finite groups $\Gamma/\ell \Gamma$ for each $ \ell \in \mathbb{N}$.