Separation for isometric group actions and hyperimaginary independence

TL;DR

This paper extends classical lemmas to isometric actions on metric spaces and applies these results to continuous logic, establishing new properties of algebraic independence and its relation to dividing independence in both discrete and continuous theories.

Contribution

It generalizes Neumann's Lemma to metric spaces and demonstrates that algebraic independence satisfies full existence and is implied by dividing independence in continuous logic.

Findings

Algebraic independence satisfies full existence.

Dividing independence implies algebraic independence.

Results are new even for discrete theories.

Abstract

We generalize P. M. Neumann's Lemma to the setting of isometric actions on metric spaces and use it to prove several results in continuous logic related to algebraic independence. In particular, we show that algebraic independence satisfies the full existence axiom (which answers a question of Goldbring) and is implied by dividing independence. We also use the relationship between hyperimaginaries and continuous imaginaries to derive further results that are new even for discrete theories. Specifically, we show that if $\mathbb{M}$ is a monster model of a discrete or continuous theory, then bounded-closure independence in $\mathbb{M}^{\text{heq}}$ satisfies full existence (which answers a question of Adler) and is implied by dividing independence.