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# An axiomatic approach to free amalgamation

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### Abstract

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. This notion encompasses 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 is NSOP4, with weak elimination of imaginaries, and use this to show that several classes of well-known homogeneous structures give new examples of (non-simple) rosy theories without the strict order property. We then prove the equivalence of simplicity and NTP2 for free amalgamation theories. As a corollary, we show that any simple free amalgamation theory, with elimination of hyperimaginaries, is 1-based. In the case of modular free amalgamation theories, we also show that simplicity coincides with NSOP3. Finally, we consider a special class of Fra\"{i}ss\'{e} limits, and prove a combinatorial characterization of simplicity, which provides new context for the fact that the generic $$K_n$$-free graphs are SOP3, while the high arity generic $$K^r_n$$-free $$r$$-hypergraphs are simple.

### Author and article information

###### Journal
2015-05-04
2015-10-01
###### Article
1505.00762