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 other than the zero-dimensional 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.
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