Higher amalgamation properties in stable theories

TL;DR

The paper constructs a canonical stable theory extension with enhanced higher independent amalgamation properties, providing explicit descriptions of finite covers and extending previous approaches.

Contribution

It introduces a method to build a stable theory $T^*$ with higher amalgamation properties from any given stable theory $T$, including explicit finite cover descriptions.

Findings

$T^*$ has higher amalgamation over algebraic closures.

Construction follows Hrushovski's approach.

Applicable to almost strongly minimal theories.

Abstract

For a complete, stable theory $T$ we construct, in a reasonably canonical way, a related stable theory $T^*$ which has higher independent amalgamation properties over the algebraic closure of the empty-set. The theory $T^*$ is an algebraic cover of $T$ and we give an explicit description of the finite covers involved in the construction of $T^*$ from $T$. This follows an approach of E. Hrushovski. If $T$ is almost strongly minimal with a $0$-definable strongly minimal set, then we show that $T^*$ has higher amalgamation over any algebraically closed subset.