Relation algebra reducts of cylindric algebras and complete representations

We show, lot any ordinal gamma >= 3. that the class RaCA gamma is pseudo-elementary and has a recursively enumerable elementary theory. S,K denotes the class of strong subalgebras of members of the class K. We devise games, F-n (3 <= n <= omega) G. H. and show, for an atomic relation algebra A with countably many atoms. that there exists has a winning strategy in F-omega (At(A)) double left right arrow A is an element of S(c)RaCA(omega), there exists has a winning strategy in F-n (At(A)) double left arrow A is an element of S(c)RaCA(n), there exists has a winning strategy in G (At(A)) double left arrow A is an element of RaCA(omega), there exists has a winning strategy in H (At(A)) double right arrow A is an element of RaCA(omega) for 3 <= n < omega. We use these games to show, for gamma >= 5 and any class K of relation algebras satisfying RaRCA(gamma) subset of K subset of S(c)RaCA(5), that K is not closed under subalgebras and is not elementary. For infinite gamma, the inclusion RaCA(gamma) subset of S(c)RaCA(7) is strict. For infinite gamma and for a countable relation algebra X we show that W has a complete representation if and only if X is atomic and 3 has a winning strategy in F(At(A)) if and only if A is atomic and A is an element of S(c)RaCA(gamma).