An infinite stratum of representability; some cylindric algebras are more representable than others

Abstract

Let $2<n<m\leq \omega$. Let $\CA_n$ denote the class of cylindric algebras of dimension $n$ and $\RCA_n$ denote the class of representable $\CA_n$s. We say that $\A\in \RCA_n$ is representable up to $m$ if $\Cm\At\A$ has an $m$-square representation. An $m$ square represenation is locally relativized represenation that is classical locally only on so called $m$-squares'. Roughly if we zoom in by a movable window to an $m$ square representation, there will become a point determinded and depending on $m$ where we mistake the $m$ square-representation for a genuine classical one. When we zoom out the non-representable part gets more exposed. For $2<n<m<l\leq \omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety $\RCA_n$ is a limiting case coinciding with $\CA_n$s having $\omega$-square representations. Let $\RCA_n^m$ be the class of algebras representable up to $m$. We show that $\RCA_n^{m+1}\subsetneq \bold \RCA_n^m$ for $m\geq n+2$.