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.