  We characterize completey (give a necessary and suffcient condition using
special neat embeddings)for a relation algebra to belong to the amalgamation,
strong amalgamation, and superamalgamation base of the class of representable
algebras. We do the same for cylindric and polyadic algebras for all dimensions
>1, infinite included. Finally, we show that we can expand our algebras by
finitely many natural operations, that force the newly expanded algebras to
have various forms of amalgamation properties, like quasi-projections and
directed cylindrifiers.
