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

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 $n\geq 3$.