Omitting types algebraically and more about amalgamation for modal cylindric algebras

Abstract

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