Some definable types that cannot be amalgamated

TL;DR

The paper investigates whether definable types necessarily admit amalgamation in all theories. It constructs a complete four-sorted theory $T$ with a levelled tree structure, and identifies definable types $p$, $q_ ext{A}$, and $q_ ext{B}$ such that no definable completion of $q_ ext{A} q_ ext{B}$ over $p$ exists, demonstrating a failure of amalgamation for definable types. This yields a theory in which global definable types do not have the amalgamation property and shows that $T$ has the independence property (IP). The result partially answers prior questions about amalgamation for definable types (in particular, about uniform definability) and highlights open directions, including whether an NIP theory can exhibit non-amalgamating definable types.

Abstract

We exhibit a theory where definable types lack the amalgamation property.

Theorems & Definitions (12)

• Definition 1.1
• Definition 1.2
• Proposition 1.3
• proof
• Lemma 1.4
• proof
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3