Notes on trace equivalence

Erik Walsberg

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arXiv · 2101.12194

Notes on trace equivalence

Erik Walsberg

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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.

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Erik Walsberg. 2021-01-28. Notes on trace equivalence. https://arxiv.org/abs/2101.12194

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