$\mathbb{Q}$ACFA

TL;DR

This paper demonstrates that many model-theoretic properties of a theory can be derived from its reducts to finite subsignatures, with applications to fields with automorphisms and their model companions.

Contribution

It establishes that properties like stability and simplicity are preserved under unions of conservative expansions, and analyzes the model companion of fields with a $(Q,+)$-action, showing it is strictly simple.

Findings

Many properties of a theory follow from properties of its reducts.

The union of stable (or simple, rosy, dependent) theories remains stable (or simple, rosy, dependent).

The model companion $Q$ACFA is strictly simple with quantifier elimination similar to ACFA.

Abstract

We show that many nice properties of a theory $T$ follow from the corresponding properties of its reducts to finite subsignatures. If $\{ T_i \}_{i \in I}$ is a directed family of conservative expansions of first-order theories and each $T_i$ is stable (respectively, simple, rosy, dependent, submodel complete, model complete, companionable), then so is the union $T := \cup_i T_i$. In most cases, (thorn)-forking in $T$ is equivalent to (thorn)-forking of algebraic closures in some $T_i$. This applies to fields with an action by $(\mathbb{Q}, +)$, whose reducts to finite subsignatures are interdefinable with the theory of fields with one automorphism. We show that the model companion $\mathbb{Q}$ACFA of this theory is strictly simple and has the same level of quantifier elimination and the same algebraic characterization of algebraic closure and forking independence as ACFA. The lattice of the fixed fields of the named automorphisms breaks supersimplicity in $\mathbb{Q}$ACFA, but away from these we find many (weakly) minimal formulas.