Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic

Abstract

Fix and let denote the class of cylindric algebras of dimension n. Roughly, is the algebraic counterpart of the proof theory of first-order logic restricted to the first n variables which we denote by . The variety of representable s reflects algebraically the semantics of . A variety of Boolean algebras with operators is atom-canonical, if whenever is atomic, then its Dedekind–MacNeille completion is also in . We show using a so-called blow up and blur construction that for any , any variety of the form containing (and including) (when ) is not atom-canonical. We show that a restricted form of that the celebrated Henkin–Orey omitting types theorem, which we refer to below as Vaught's Theorem (), fails dramatically for even if we allow only so-called m-square models for . We deduce that many multi-modal logics, like , the m clique guarded fragments and packed fragments of are not Sahlqvist. In contrast, we prove a positive for theories with respect to usual semantics, by imposing extra conditions (such as quantifier elimination) on possibly uncountable theories considered and/or the non-principal types omitted (that are also allowed to be possibly uncountable) such as completeness.ast is a maximality condition delineating the edge of an independent statement to a provable one. is shown that the maximality condition cannot be removed even to prove the weaker for uncountable theories; and this construction is used to reprove a celebrated result of Hirsch and Hodkinson, namely, that the class of completely representable s is not el.