The free non-commutative cylindric algebras are not atomic

Abstract

The founding of the theory of cylindric algebras, by Alfred Tarski, was a conscious effort to create algebras out of first order predicate calculus. Let $n\in\omega$. The classes of non-commutative cylindric algebras ($NCA_n$) and weakened cylindric algebras ($WCA_n$) were shown, by Istv\'an N\'emeti, to be examples of decidable versions of first order logic with $n$ variables. In this article, we give new proofs for the decidability of the equational theories of these classes. We also give an answer to the open problem, posed by N\'emeti in 1985, addressing the atomicity of the finitely generated free algebras of these classes. We prove that all the finitely generated free algebras of the varieties $NCA_n$ and $WCA_n$ are not atomic. In other words, we prove that the corresponding versions of first order logic have weak G\"odel's incompleteness property.