Algebra universalis

, 80:41 | Cite as

Atoms in infinite dimensional free sequence-set algebras

  • Mohamed KhaledEmail author
  • István Németi


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.


Free algebras Cylindric-like algebras Atoms and zero-dimensional elements 

Mathematics Subject Classification

03G15 03B20 06E25 



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Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesBahçeşehir UniversityIstanbulTurkey
  2. 2.Alfréd Rényi Institute of MathematicsBudapestHungary

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