Complete Representations and Neat Embeddings

Abstract

Let $2<n<\omega$. Then ${\mathsf{C}\mathsf{A}}_n$ denotes the class of cylindric algebras of dimension $n$, ${\mathsf{R}\mathsf{C}\mathsf{A}}_n$ denotes the class of representable $\mathsf{C}{\mathsf{A}}_{\mathsf{n}}$s, ${\mathsf{C}\mathsf{R}\mathsf{C}\mathsf{A}}_n$ denotes the class of completely representable ${\mathsf{C}\mathsf{A}}_n$s, and ${\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{\omega }(\subseteq {\mathsf{C}\mathsf{A}}_n$) denotes the class of $n$-neat reducts of ${\mathsf{C}\mathsf{A}}_{\omega }$s. The elementary closure of the class ${\mathsf{C}\mathsf{R}\mathsf{C}\mathsf{A}}_n$s (${\mathbf{K}}_{\mathbf{n}}$) and the non-elementary class $\mathsf{A}\mathsf{t}({\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{\omega })$ are characterized using two-player zero-sum games, where $\mathsf{A}\mathsf{t}$ is the operator of forming atom structures. It is shown that ${\mathbf{K}}_{\mathbf{n}}$ is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ${\mathbf{S}}_{\mathbf{c}}{\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{\omega }$ where ${\mathbf{S}}_{\mathbf{c}}$ is the operation of forming complete subalgebras. For any class $\mathbf{L}$ such that $\mathsf{A}\mathsf{t}{\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{\omega }\subseteq \mathbf{L}\subseteq \mathsf{A}\mathsf{t}{\mathbf{K}}_{\mathbf{n}}$, it is proved that $\mathbf{S}\mathbf{P}\mathfrak{Cm}\mathbf{L}={\mathsf{R}\mathsf{C}\mathsf{A}}_n$, where $\mathsf{C}\mathsf{m}$ is the dual operator to $\mathsf{A}\mathsf{t}$; that of forming complex algebras. It is also shown that any class $\mathbf{K}$ between ${\mathsf{C}\mathsf{R}\mathsf{C}\mathsf{A}}_n\cap {\mathbf{S}}_{\mathbf{d}}{\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{\omega }$ and ${\mathbf{S}}_{\mathbf{c}}{\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_{n+3}$ is not first order definable, where ${\mathbf{S}}_{\mathbf{d}}$ is the operation of forming dense subalgebras, and that for any $2<n<m$, any $l\ge n+3$ any any class $\mathbf{K}$ (such that $\mathsf{A}\mathsf{t}({\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_m)\cap {\mathsf{C}\mathsf{R}\mathsf{C}\mathsf{A}}_n\subseteq \mathbf{K}\subseteq \mathsf{A}\mathsf{t}{\mathbf{S}}_{\mathbf{c}}{\mathsf{N}\mathsf{r}}_n{\mathsf{C}\mathsf{A}}_l$, $\mathbf{K}$ is not not first order definable either.