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

Fix 2 < n < ω. Ln denotes first order logic restricted to the first n variables and for any ordinals α < β, (R)CAα denotes the class of (representable) cylindric algebras of dimension α, and NrαCAβ denotes the class of α-neat reducts of CAβ . Certain CAns constructed from relation algebras having an n–dimensional cylindric basis are used to show that Vaught’s Theorem (VT) looked upon as a special case of the omitting types theorem (OTT) fails in the m-clique guarded fragment (CGFm) of Ln, when m ≥ n + 3. For infinitely many values of n ≤ l < m ≤ ω, there is an atomic, countable and complete Ln theory T such that the type of co-atoms (of the formula algebra FmT ) is realizable in every m-square model of T but cannot be isolated using l variables. Here ‘m-squareness’ is the locally well behaved clique-guarded semantics of CGFm; an m-square model is l-square, but the converse may be false. The limiting case, an ω-square model, is an ordinary model. This is proved algebraically by constructing a countable, atomic and simple algebra A ∈ RCAn ∩ NrnCAl whose Dedekind-MacNeille completion (CmAtA) does not have an m-square representation, a fortiorti CmAtA / ∈ SNrnCAm(⊇ RCAn). OTTs are proved with respect to standard semantics for Ln countable theories that have quantifier elimination; it is shown that < 2 many non-principal types can be omitted in case they are maximal. Our purpose throughout the paper is twofold. Apart from presenting novel ideas of applying algebra to logic, we present our new results in both algebraic and modal logic in an integrated format. Fix 2 < n < ω. We use blow up and blur constructions to proving non-atom canonicity of several varities of relation and cylindric algebras. We recall that a class K of Boolean algebras with operators (BAOs) is atom–canonical if whenever A ∈ K with atom structure AtA is completey additive, then its Dedekind-MacNeille completion, namely, the complex algebra of its atom structure CmAtA is also in K. This subtle construction may be applied to any two classes L ⊆ K of completely additive BAOs. One takes an atomic A / ∈ K (usually but not always finite), blows it up, by splitting one or more of its atoms each to infinitely many subatoms, obtaining an (infinite) countable atomic Bb(A) ∈ L, such that A is blurred in Bb(A) meaning that A does not embed in Bb(A), but A embeds in the Dedekind-MacNeille completion of Bb(A), namely, CmAtBb(A). Then any class M say, between L and K that is closed under forming subalgebras will not be atom–canonical, for Bb(A) ∈ L(⊆ M), but CmAtBb(A) / ∈ K(⊇ M) because A / ∈ M and SM = M. We say, in this case, that L is not atom–canonical with respect to