Atoms in infinite dimensional free sequence-set algebras

Khaled M. K. M. I., Németi I.

ALGEBRA UNIVERSALIS, vol.80, no.4, 2019 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 80 Issue: 4
  • Publication Date: 2019
  • Doi Number: 10.1007/s00012-019-0610-8
  • Journal Name: ALGEBRA UNIVERSALIS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Free algebras, Cylindric-like algebras, Atoms and zero-dimensional elements
  • Istanbul Medipol University Affiliated: No

Abstract

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