  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.
