Non elementary classes of relation and cylindric algebras

Abstract

For any pair of ordinals $\alpha<\beta$, $\sf CA_\alpha$ denotes the class of cylindric algebras of dimension $\alpha$, $\sf RCA_{\alpha}$ denote the class of representable $\sf CA_\alpha$s and $\sf Nr_\alpha CA_\beta$ ($\sf Ra CA_\beta)$ denotes the class of $\alpha$-neat reducts (relation algebra reducts) of $\sf CA_\beta$. We show that any class $\sf K$ such that $\sf RaCA_\omega \subseteq \sf K\subseteq RaCA_5$, $\sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<\omega$. It is also shown that any class $\sf K$ such that $\sf Nr_nCA_\omega \cap {\sf CRCA}_n\subseteq {\sf K}\subseteq \mathbf{S}_c\sf Nr_nCA_{n+3}$, where $\sf CRCA_n$ is the class of completely representable $\sf CA_n$s, and $\mathbf{S}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $\sf K$ such that $\mathbf{S}_d\sf Ra CA_\omega \subseteq {\sf K}\subseteq \mathbf{S}_c\sf RaCA_5$ is not elementary. It remains to be seen whether there exist elementary classes between $\sf RaCA_\omega$ and $\mathbf{S}_d\sf RCA_{\omega}$. In particular, for $m\geq n+3$, the classes $\sf Nr_nCA_m$, $\sf CRCA_n$, $\mathbf{S}_d\sf Nr_nCA_m$, where $\mathbf{S}_d$ is the operation of forming dense subalgebras are not first order definable.