Completions, Complete Representations and Omitting Types

Abstract

Algebraic logic arose as a subdiscipline of algebra mirroring constructions and theorems of mathematical logic. It is similar in this respect to such fields as algebraic geometry and algebraic topology, where the main constructions and theorems are algebraic in nature, but the main intuitions underlying them are respectively geometric and topological. The main intuitions underlying algebraic logic are, of course, those of formal logic. Investigations in algebraic logic can proceed in two conceptually different, but often (and unexpectedly) closely related ways. First one tries to investigate the algebraic essence of constructions and results in logic, in the hope of gaining more insight that one could add to his understanding, thus to his knowledge. Second, one can study certain “particular” algebraic structures (or simply algebras) that arise in the course of his first kind of investigations as objects of interest in their own right and go on to discuss questions which naturally arise independently of any connection with logic. But often such purely algebraic results have impact on the logic side. Examples are the undecidability of the representation problem for finite relation algebras [Hir-Hod,01d] that led to deep results concerning undecidability of product modal logics [Hir-Hod-Kur,02a] answering problems of Gabbay and Shehtman.