Martin's axiom, omitting types, and complete representations in algebraic logic

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 < n ≤ ω via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin's axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey's omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 < n < ω. Ln has been recently (and quite extensively) studied as a many-dimensional modal logic. © 2002 Kluwer Academic Publishers.