Not All Representable Cylindric Algebras Are Neat Reducts

Cylindric algebras are the algebraic counterparts of First Order Logic as was explained in the monograph [1] of Henkin, Monk, and Tarski, and also in [2], [3], and [4]. A cylindric algebra is representable if it corresponds to some logical system in a strong sense, cf. Theorem 4.2 and Definition 6.2 in [2] and 1.1.13 of [1]. (see also the remark preceding Corollary 2 in the present note). It was shown in [1], cf. Corollary 3.14 and Corollary 3.18 of [2], that the class Rα od all representable cylindric algebras of dimension α coincides with the class S Nrα CAα+ω of all subalgebras of neat reducts. Here NrαCAα+ω denotes the class of all neat reducts, see 2.6.28 of [1]. Therefore neat reducts are strongly related to algebraic versions of logical systems, cf. 2.6.26 of [1]. (See the remarks preceding Corollary 2 in the present note). The question arose how close this relation is: Problem 2.11 on p. 464 of [1] is the question whether the class NrαCAβ of all α-dimensional neat reducts of β-dimensional cylindric algebras is closed under the formation of subalgebras and homomorphic images or not. The Theorem below formulates an answer to this question. We shall use the notations of [1], e.g. if K is a class of algebras then S K and H K are the classes of all subalgebras of elements of K and all homomorphic images of elements of K, respectively.