DRAFT 1.3, Dec. 1996. Not guaranteed error-free.
Relative to earlier drafts, this version has some errors
corrected and the presentation slightly simplified. Comments are welcome.
For any finite n (at least 3), there are two atomic n-dimensional cylindric algebras
with the same atom structure, with one representable, the other, not.
Hence, the complex algebra of the atom structure of a representable cylindric algebra
is not always representable, so that the class RCA_n of representable n-dimensional
cylindric algebras is not closed under completions. This answers a question of Monk.
Further, it follows by an argument of Venema that RCA_n is not axiomatisable by Sahlqvist
equations, nor by equations where negation can only occur in constant terms. This
answers a question of Henkin, Monk, and Tarski.
Here is a ps.gz
file of the paper. 21 pages.