Model theoretic dynamics in Galois fashion

TL;DR

This paper explores the model theory of structures with group actions, showing that under certain conditions, these models are simple, have quantifier elimination properties similar to ACFA, and possess geometric elimination of imaginaries.

Contribution

It introduces a framework for analyzing existentially closed models with group actions, establishing their simplicity and elimination properties under boundedness assumptions.

Findings

Models are simple and eliminate quantifiers up to existential formulas.

They code finite sets and have geometric elimination of imaginaries.

Not all models have weak elimination of imaginaries.

Abstract

We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D. Assuming that the class of these existentially closed models is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas - similar to ACFA. Moreover, it codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.