An axiomatic approach to free amalgamation

TL;DR

This paper introduces free amalgamation theories via axioms of abstract ternary relations, showing they lack the strict order property, have elimination of hyperimaginaries, and connect to well-known homogeneous structures, providing new insights into simplicity and related properties.

Contribution

It defines free amalgamation theories using axioms, establishes their model-theoretic properties, and characterizes simplicity in Fraisse limits with free amalgamation, linking to known structures.

Findings

Free amalgamation theories lack the strict order property.

They have elimination of hyperimaginaries and weak elimination of imaginaries.

Simplicity coincides with NTP2 and, under modularity, with NSOP3.

Abstract

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraisse limits with free amalgamation, which provides new context for the fact that the generic $K_n$-free graphs are SOP3, while the higher arity generic $K^r_n$-free $r$-hypergraphs are simple.