The strong amalgamation property into union

Paolo Lipparini

Published 2021-02-28Version 1

We consider the situation in which some class of structures has the Strong Amalgamation Property (SAP) with the further requirement that the amalgamating structure can be taken over the set theoretical union of (the images of) the domains of the structures to be amalgamated. We call this property SAPU. The main advantage of SAPU over SAP is that there are many preservation theorems showing that we can merge different theories with SAPU still obtaining a theory with SAPU, hence with SAP. In particular, we elaborate on some classical results by R. Fra\"iss\'e and B. J\'onsson, showing that the class of partially ordered sets with a family of operators has SAPU. More generally, the argument applies to various theories with possibly many binary relations and unary operations together, possibly satisfying some coarseness, isotonicity and closure conditions.