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arXiv · 1903.01527

Atoms in infinite dimensional free sequence-set algebras

Mohamed Khaled·István Németi
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Abstract

A. Tarski proved that the m-generated free algebra of $\mathrm{CA}_α$, the class of cylindric algebras of dimension $α$, contains exactly $2^m$ zero-dimensional atoms, when $m\ge 1$ is a finite cardinal and $α$ is an arbitrary ordinal. He conjectured that, when $α$ 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 $\mathrm{CA}_α$ is replaced by $\mathrm{D}_α$, $\mathrm{G}_α$, but the $m$-generated free $\mathrm{Crs}_α$ algebra is atomless.

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Mohamed Khaled, István Németi. 2019-03-04. Atoms in infinite dimensional free sequence-set algebras. https://arxiv.org/abs/1903.01527

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