A remark on strict independence relations

TL;DR

The paper establishes a correspondence between strict independence relations in a complete theory with weak elimination of imaginaries and its expansion, and demonstrates that for a specific Fraïssé limit theory, multiple strict independence relations exist, answering an open question.

Contribution

It provides an explicit bijection between independence relations in a theory and its imaginary expansion, and shows the existence of multiple such relations in a particular Fraïssé limit theory.

Findings

Bijection between independence relations in T and T^{eq}

Existence of multiple strict independence relations in the Fraïssé limit theory of finite metric spaces

Answers an open question by Adler about independence relations in this context

Abstract

We prove that if $T$ is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for $T$ and strict independence relations for $T^{\text{eq}}$. We use this observation to show that if $T$ is the theory of the Fra\"{i}ss\'{e} limit of finite metric spaces with integer distances, then $T^{\text{eq}}$ has more than one strict independence relation. This answers a question of Adler [1, Question 1.7].