Atom canonicity and first order definability in classes of algebras of relations

Abstract

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn 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 CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω 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 RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever A ∈ 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 HenkinOrey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that SdNrnCAω ∩ CRCAn ⊆ K ⊆ ScNrnCAn+3, where CRCAn is the class of completely representable CAns, and Sd(Sc) denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω ⊆ K ⊆ ScRaCA5 is not elementary.