Omitting types algebraically and more about amalgamation for modal cylindric algebras.

Let α be an arbitrary infinite ordinal, and 2<n<ω. In [26] we studied—using algebraic logic—interpolation and amalgamation for an extension of first order logic, call it Lα, with α many variables, using a modal operator of a unimodal logic L that contributes to the semantics. Our algebraic apparatus was the class of modal cylindric algebras. Modal cylindric algebras, briefly LCAα, are cylindric algebras of dimension α, expanded with unary modalities inheriting their semantics from a unimodal logic L such as K5, S4 or S5. When L=S5 modal cylindric algebras based on L are just cylindric algebras, that is to say, S5CAα=CAα. This paper is a sequel to [26], where we study algebraically other properties of Lα. We study completeness and omitting types (OTTs) for Lαs by proving several representability results for so‐called dimension complemented and locally finite LCAα. Furthermore, we study the notion of atom‐canonicity for LCAn, the variety of n‐dimensional modal cylindric algebras. Atom canonicity, a well known persistence property in modal logic, is studied in connection to OTT for Ln, which is Lω restricted to the first n variables. We further continue our study of interpolation in [26] for algebraizable extensions of Lα by studying LCAα using both algebraic logic and category theory. Our main results on OTT are Theorems 3.7, 4.4 & 4.6, while our main results on amalgamation are Theorems 5.7, 5.10, 5.13 & 5.16. [ABSTRACT FROM AUTHOR]