On neat embeddings of cylindric algebras

We show that certain properties of dimension complemented cylindric algebras, concerning neat embeddings, do not generalize much further. Let α ≥ ω. There are non‐isomorphic representable cylindric algebras of dimension α each of which is a generating subreduct of the same β dimensional cylindric algebra. We also show that there exists a representable cylindric algebra ?? of dimension α, such that ?? is a generating subreduct of ?? and ??′, both in CA_{α +ω} , however ?? and ??′ are not isomorphic. This settle questions raised by Henkin, Monk and Tarski (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)