Atoms in infinite dimensional free sequence-set algebras

Abstract

A. Tarski proved that the m-generated free algebra of ${\mathrm{CA}}_{\alpha }$, the class of cylindric algebras of dimension $\alpha$, contains exactly $2^m$ zero-dimensional atoms, when $m\ge 1$ is a finite cardinal and $\alpha$ is an arbitrary ordinal. He conjectured that, when $\alpha$ is infinite, there are no more atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski's conjecture is true if ${\mathrm{CA}}_{\alpha }$ is replaced by ${\mathrm{D}}_{\alpha }$, ${\mathrm{G}}_{\alpha }$, but the $m$-generated free ${\mathrm{Crs}}_{\alpha }$ algebra is atomless.