The free non-commutative cylindric algebras are not atomic

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Khaled M. K. M. I.

Logic Journal of the IGPL, vol.25, no.5, pp.673-685, 2017 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 25 Issue: 5
  • Publication Date: 2017
  • Doi Number: 10.1093/jigpal/jzw058
  • Journal Name: Logic Journal of the IGPL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.673-685
  • Keywords: Algebraic logic, Decidability, Disjunctive normal forms, Gödel's incompleteness properties, Non-atomicity of the free algebras, Non-commutative cylindric algebras, Non-permutable first-order logic
  • Istanbul Medipol University Affiliated: No


The classes of non-commutative cylindric algebras and weakened cylindric algebras were shown, by István Németi, to have decidable equational theories. In this article, we give new proof for this result and we give an answer to the open problem, posed by Németin 1985, addressing the atomicity of the finitely generated free algebras of these classes. We prove that the free algebras of these classes are not atomic.