Higher amalgamation in $\mathrm{ACFA}^{+}$

TL;DR

This paper investigates higher amalgamation properties in the theory of difference fields with an additive character, revealing limitations at the 4-amalgamation level and conditions for full n-amalgamation over certain substructures.

Contribution

It demonstrates that the known 3-amalgamation condition is insufficient for 4-amalgamation in $ ext{ACFA}^+$ and establishes that full n-amalgamation holds over specific substructures.

Findings

3-amalgamation condition is not sufficient for 4-amalgamation

Full n-amalgamation holds over substructures with $ ext{ACFA}$ reducts

Identifies limitations and conditions for higher amalgamation in $ ext{ACFA}^+$

Abstract

We show two results on higher amalgamation in the theory $\mathrm{ACFA}^{+}$, the model companion of the theory of difference fields with an additive character (added as a continuous logic predicate) on the fixed field in characteristic 0. On one hand, we show that the non-trivial condition for 3-amalgamation established in a preceding paper is not sufficient for 4-amalgamation. On the other hand, we show that when working over substructures whose $\mathcal{L}_{\sigma}$-reduct is a model of $\mathrm{ACFA}$, $n$-amalgamation holds for all $n\geq 3$.