  We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is
pseudo elementary. When n and m are finite we use a two sorted theory, when n
is finite and m infinite we use a three sorted one, and finally when both are
infinite we use a four sorted defining theory. Our non finite axiomatizability
result, follows from the fact that for 2<n<m, and any r\in \omega there exists
a finite (Monk like) algebra C(m,n,r), such that C(m,n,r)\in Nr_nCA_m
C(m,n,r)\notin SNr_nCA_{m+1}, and any non trivial ultraproduct on r of such
algebras in in ElNr_nCA_m. Finally we use such algebras, to show that for
infinite dimension there is an algebra A\in Nr_{\alpha}\CA_{\apha+k} that is
not in SNr_{\alpha}CA_{\alpha+k+1} (\alpha an infinite ordinal).
