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 alpha >= omega. There are non-isomorphic representable cylindric algebras of dimension alpha each of which is a generating subreduct of the same beta dimensional cylindric algebra. We also show that there exists a representable cylindric algebra U of dimension alpha, such that 21 is a generating subreduct of b and B', both in CA(alpha+omega) however B and B' are not isomorphic. This settle questions raised by Henkin, Monk and Tarski. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim