Transfer of generalized amalgamation in simple theories

We give an abstract framework to transfer generalized amalgamation from a simple theory to another, and we apply it to theories of lovely pairs and of bounded PAC structures. We show in particular that bounded pseudo-algebraically closed fields have generalized amalgamation, regardless of their imperfection degree.

💡 Research Summary

The paper develops an abstract mechanism for transferring the property of generalized n‑amalgamation from one simple theory to another and applies this mechanism to several important model‑theoretic contexts, notably lovely pairs and bounded pseudo‑algebraically closed (PAC) structures.

The authors begin by recalling the Kim‑Pillay characterization of simplicity via a ternary non‑forking independence relation |⊥| and the associated independence theorem, which can be reformulated as a 3‑amalgamation property. They then introduce the notion of a W‑amalgamation system over a parameter set C: for each finite subset w of a fixed index set, a type p_w(x_w) is assigned satisfying consistency, algebraic closure, and independence conditions. When such a system can be completed (i.e., extended to a type over the union of all variables), the theory is said to have W‑amalgamation; if this holds for all W closed under subsets of P(