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It is shown that not every Boolean algebra is the Boolean part of a nondiscrete relation or cylindric algebra, but every nonatomless Boolean algebra is. Solutions of Tarski's Cube Problem for nondiscrete relation and cylindric algebras are given. Introduction. Relation algebras (RAs) and cylindric algebras (CAs) are Boolean algebras (BAs) endowed with additional structure. The question naturally arises: Which BAs are Boolean parts (i.e. Boolean reducts) of RAs or CAs? This question can be considered as a representation problem for BAs, too: Which BA can be represented as a BA of some binary relations on a set U closed under composition, inversion, and containing the identity relation on U (this is the RA case); and which BA can be represented as a BA of n-ary relations closed under, roughly, first-order definability, i.e. containing with every relation Rj,..., Rk the relations first-order definable (with n variables) from them, too? RA and CA,a denote the classes of RAs and CAs of dimension a respectively. There is a trivial way of turning every BA into a RA or a CAa; these RAs and CAs are called "Boolean" and "discrete" ones (cf. [7, p. 276 and 5, 1.3.11]). By [5, 1.2.14], every BA is the Boolean part of a nondiscrete CA1. J. D. Monk in 1981 and independently R. Laver in 1984 asked the question: Which BAs are Boolean parts (reducts) of nondiscrete CA ,s, a( > 2? Theorems 1 and 2 give partial answers. To CAa5s there correspond another BA, the so-called zero-dimensional part 3b W of a CA, W. By [5, 2.4.35], every BA is the zero-dimensional part of a hereditarily nondiscrete CA a (for all a > 0). In the last part of the paper we use this fact (together with Ketonen's solution for BAs) to give a full solution of Tarski's Cube Problem for CA,as as well as for RAs (of course we mean nondiscrete CA,as and RAs here). See Theorems 3 and 4. This solves that part of Problem 2.4 of [5] which was still left open in [6, p. 127]. The definitions of RA and CA.a, both originating with Tarski, can be found e.g. in [5, 7, 8, 10] and in [5,6] respectively. Below we recall that part of their definitions that will be needed in the present paper. Received by the editors April 2, 1985 and, in revised form, August 19, 1985 and February 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03G05, 03G25; Secondary 03E15, 06E99. C1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Boolean reducts of relation and cylindric algebras and the cube problem