Homogenizable relational structures
A class \({\mathcal C}\) of structures has the local failure of amalgamation iff there are in \({\mathcal C}\) finitely many diagrams \({\mathcal D}_ j\) such that any diagram \({\mathcal D}\) in \({\mathcal C}\) fails to be amalgamated in \({\mathcal C}\) iff one of the \({\mathcal D}_ j's\) is embeddable in \({\mathcal D}\). Then the author proves: Let \({\mathcal L}\) be a purely relational languages. Let \(\Sigma\) be a set of universal \({\mathcal L}\)-sentences. Let \({\mathcal C}\) be the class of models of \(\Sigma\). Assume that the class of finite \({\mathcal C}\)-structures satisfies the Joint Embedding Property and that it has Local Failure of Amalgamation. Then \({\mathcal C}\) is homogenizable. It is shown further that the theory of these ``homogenized'' structures is model complete.
scientific article; zbMATH DE number 4158612