Notes on trace equivalence

TL;DR

The paper introduces trace definability and trace equivalence as weak notions of interpretability and equivalence for first-order structures, with applications to NIP theories and expansions of ordered groups.

Contribution

It develops the concepts of trace definability and trace equivalence, connecting them to indiscernible collapse and classifying expansions of ordered groups based on trace properties.

Findings

Trace equivalence provides a new way to compare first-order structures.

For certain expansions of ordered groups, exactly one of two trace-related conditions holds.

The work links trace notions to interpretability and definability in model theory.

Abstract

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we get an interesting weak equivalence notion for $\mathrm{NIP}$ theories. We describe a close connection to indiscernible collapse. We also show that if $Q$ is a divisible subgroup of $(\mathbb{R};+)$ and $\mathcal{Q}$ is a dp-rank one expansion of $(Q;+,<)$ then exactly one of the following holds: $\mathrm{Th}(\mathcal{Q})$ trace defines $\mathrm{RCF}$ or $\mathcal{Q}$ is trace equivalent to a reduct of an ordered vector space.