Abstract
Fix 2 < n < omega and let CA(n) denote the class of cyindric algebras of dimension n. Roughly CA(n) is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by L-n. The variety RCA(n) of representable CA(n)s reflects algebraically the semantics of L-n. Members of RCA(n) are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CA(n) has a finite equational axiomatization, RCA(n) is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CA(n) substantially richer than that of Boolean algebras, just as much as L omega,omega is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCA(n) are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever A is an element of V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of L omega,omega, fails dramatically for L-n even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that Nr(n)CA omega boolean AND CRCA(n) subset of K subset of S(c)Nr(n)CA(n+3), where CRCA(n) is the class of completely representable CA(n)s, and S-c denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that S(d)R(a)CA omega subset of K subset of S(c)R(a)CA(5) is not elementary, where S-d denotes the operation of forming dense subalgebra.