A single schema notion for systems of varieties of Boolean algebras with operators that covers both Monk's cylindric schemas and Halmos' polyadic schemas, and that integrates the finite dimensions into the infinite-dimensional definition. Because a schema instantiated in every dimension is preserved by the ultraproduct 'stretching' argument, finite-dimensional facts - non-finite-axiomatizability of the representable algebras, failure of the neat embedding hierarchy to collapse, non-atom-canonicity - lift verbatim to transfinite dimensions, and amalgamation and super-amalgamation can be settled uniformly for whole families of cylindric-like varieties at once, including MV polyadic algebras, reducts of Heyting polyadic algebras, and Ferenczi's cylindric-polyadic algebras. The same schema view recasts neat reducts as a functor whose right adjoint exists exactly when the target class amalgamates.
This arXiv paper by Tarek Sayed Ahmed provides a comprehensive treatment of cylindric and polyadic algebras with novel approaches. It generalizes Monk's schema to integrate finite dimensions, enablin…