  The classes of relativized relation algebras (whose units are not necessarily
transitive as binary relations) are known to be finitely axiomatizable. In this
article, we give a new proof for this fact that is easier and more transparent
than the original proofs. We give direct constructions for all cases, whereas
the original proofs reduced the problem to only one case. The proof herein is
combinatorial and it uses some techniques from game theory.
