PAC structures as invariants of finite group actions

TL;DR

This paper explores the model theory of finite group actions on stable structures, characterizing existentially closed actions via invariants and PAC structures, and establishing conditions for the existence of a model companion.

Contribution

It provides an abstract description of existentially closed actions using invariants and PAC structures, and proves the PAC property is first order in several theories of interest.

Findings

Existentially closed actions characterized by invariants and PAC structures.

PAC property is first order in various theories of fields of positive characteristic.

Conditions established for the existence of a model companion for these actions.

Abstract

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.