  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 get SAPU for various
theories with many binary relations, each relation satisfying any set of
properties chosen among transitivity, reflexivity, symmetry, antireflexivity,
antisymmetry. We may also add unary operations, possibly satisfying some
coarseness, isotonicity and closure conditions.
  SAPU is not limited to relational theories: the varieties defining the most
usual Maltsev conditions in universal algebra have SAPU. Other examples include
bounded directoids, order algebras and various generalization.
