Paper detail

The elementary closure of the class Nr_nCA_m for m\geq n+1 is not finitely axiomatizable, futhermore for any finite k\geq 1, there is A\in Nr_ωCA_{\omeg+k}that is not SNr_ωCA_{ω+k+1}

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 ωthere 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_α\CA_{\apha+k} that is not in SNr_αCA_{α+k+1} (αan infinite ordinal).