As is well-known, a classical representation theorem of the theory of cylindric algebras is: Aε IGwsaif and only ifA ε SNrαCAα+ε.The part “only if” is trivial. Regarding to the other part“A ε SNrαCAα+εthen Aε IGwsα“the following question arises: is it possible to replace the class CA in the hypothesis Aε SNrαCAα+εby a larger class so that the theorem still holds. Such a larger class Kαβis defined. The class Kαβis the best possible, in a sense to be made precise. Representability of set algebras is investigated, too.