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Atoms in infinite dimensional free sequence-set algebras

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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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Authors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Bahçeşehir University, Istanbul, Turkey

    Mohamed Khaled

  2. Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    István Németi

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  1. Mohamed Khaled
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  2. István Németi
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Correspondence to Mohamed Khaled.

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Khaled, M., Németi, I. Atoms in infinite dimensional free sequence-set algebras. Algebra Univers. 80, 41 (2019). https://doi.org/10.1007/s00012-019-0610-8

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