Non elementary classes of relation and cylindric algebras

Abstract

For any pair of ordinals $\alpha <\beta$, $\mathsf{C}{\mathsf{A}}_{\alpha }$ denotes the class of cylindric algebras of dimension $\alpha$, $\mathsf{RC}{\mathsf{A}}_{\alpha }$ denote the class of representable $\mathsf{C}{\mathsf{A}}_{\alpha }$s and $\mathsf{N}{\mathsf{r}}_{\alpha }\mathsf{C}{\mathsf{A}}_{\beta }$ ($\mathsf{RaC}{\mathsf{A}}_{\beta })$ denotes the class of $\alpha$-neat reducts (relation algebra reducts) of $\mathsf{C}{\mathsf{A}}_{\beta }$. We show that any class $\mathsf{K}$ such that $\mathsf{RaC}{\mathsf{A}}_{\omega }\subseteq \mathsf{K}\subseteq \mathsf{RaC}{\mathsf{A}}_5$, $\mathsf{K}$ is not elementary, i.e not definable in first order logic. Let $2<n<\omega$. It is also shown that any class $\mathsf{K}$ such that $\mathsf{N}{\mathsf{r}}_{\mathsf{n}}\mathsf{C}{\mathsf{A}}_{\omega }\cap {\mathsf{CRCA}}_{\mathsf{n}}\subseteq \mathsf{K}\subseteq {\mathbf{S}}_{\mathsf{c}}\mathsf{N}{\mathsf{r}}_{\mathsf{n}}\mathsf{C}{\mathsf{A}}_{\mathsf{n}+3}$, where $\mathsf{CRC}{\mathsf{A}}_{\mathsf{n}}$ is the class of completely representable $\mathsf{C}{\mathsf{A}}_{\mathsf{n}}$s, and ${\mathbf{S}}_c$ denotes the operation of forming complete subalgebras, is proved not to be elementary. Finally, we show that any class $\mathsf{K}$ such that ${\mathbf{S}}_d\mathsf{RaC}{\mathsf{A}}_{\omega }\subseteq \mathsf{K}\subseteq {\mathbf{S}}_{\mathsf{c}}\mathsf{RaC}{\mathsf{A}}_5$ is not elementary. It remains to be seen whether there exist elementary classes between $\mathsf{RaC}{\mathsf{A}}_{\omega }$ and ${\mathbf{S}}_d\mathsf{RC}{\mathsf{A}}_{\omega }$. In particular, for $m\ge n+3$, the classes $\mathsf{N}{\mathsf{r}}_{\mathsf{n}}\mathsf{C}{\mathsf{A}}_{\mathsf{m}}$, $\mathsf{CRC}{\mathsf{A}}_{\mathsf{n}}$, ${\mathbf{S}}_d\mathsf{N}{\mathsf{r}}_{\mathsf{n}}\mathsf{C}{\mathsf{A}}_{\mathsf{m}}$, where ${\mathbf{S}}_d$ is the operation of forming dense subalgebras are not first order definable.

arXiv admin note: text overlap with arXiv:1503.02189, arXiv:1408.3282, arXiv:1608.03513, arXiv:1502.07701