Atom-canonicity, relativized representations and omitting types for clique guarded semantics and guarded logics

Abstract

We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting types theorems, both negative and positive for finite variable fragments and / or modifications of first order logic. Negative results are obtained when we keep usual syntax and relativize models (so that they witness commutativity of quantifiers only locally) and positive ones are obtained when we weaken 'commutativity of quantifiers' in the syntax and relativize semantics differently. Such algebras have weak neat embedding properties, too, in the sense that they embed into neat reducts of algebras in higher dimensions possibly finite. In the second part of the paper various notions of representability originally formulated for atom structure are lifted in an obvious way to the algebra level, like weak and strong representability. Such classes of algebras (that are atomic) are characterized completely via neat embeddings. Finally, several model theoretic questions on such classes consisting of algebras having weak neat embedding properties and relativized representations, like decidability of their equational or universal theory, their finite axiomatizability if first order definable, are posed and answered.