© 2000 by Oxford University Press
An introduction to cylindric set algebras
Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, CO 80309-0395, USA E-mail: monkd@euclid.colorado.edu
These notes are a corrected and revised version of notes which accompanied lectures given at the Banach Center in the fall of 1991. The intent is to give a self-contained introduction to cylindric algebras from the concrete point of view. I hope that after reading this introduction the reader will be able to digest the basic works on this subject (Henkin, Monk, Tarski [19,20] and Henkin, Monk, Tarski, Andréka, Németi [21], which appears in a condensed form in Henkin, Monk, Tarski [20]. One of the frightening things about both of these books is that they begin with a mass of definitions and proceed with very detailed discussion of the interrelationships of the defined notions. We are going to introduce just a few of these definitions, little by little, giving important (but not highly technical) results about them as we go along. And we will try to motivate the notions from logic. See the end of these notes for indices of symbols and words.
Cylindric algebras form the most developed form of algebraic logic. In general, algebraic logic is concerned with algebraic structures which correspond to logics of various sorts. Cylindric algebras correspond to ordinary first order logics and to certain straightforward modifications of these logics. Other algebraic structures have a similar relationship to first order logic; the most developed of these are relation algebras (in Tarski's sense) and polyadic algebras. We will not be concerned with these, but the reader should be able to study them more easily after reading these notes.
We will describe only the concrete aspect of cylindric algebras. The axiomatic version, fully developed in Henkin, Monk, Tarski [19], will play only a minor role. Also, we will not deal with applications. Such applications exist in several other fields, such as combinatorics and theoretical computer science.
We assume familiarity with the elementary theory of Boolean algebras, elementary first order logic, and with the basics of universal algebra.