Neatly atomic cylindric algebras and isomorphisms

The result proved in this note, which is related closely to some results of Vaught on atomic theories, is the following model theoretical generalization of the characterization of the atomic Boolean algebras with their isomorphism (viz. that any Boolean algebra A is atomic iff every isomorphism from A onto a Boolean algebra is a lower base isomorphism): if T is a complete theory in a countable language, then the following conditions are equivalent: (i) T is atomic
(ii) Any two models of T have isomorphic submodels (iii) T has a countable model A such that A and any model of T have isomorphic submodels. This theorem can be formulated in cylindric set algebraic terms as follows: If A is a countably generated infinite dimensional locally finite regular cylindric set algebra with a countable base, then the neat n-reduct of A is atomic for any finite n iff any isomorphism from A onto a regular cylindric set algebra is a lower base-isomorphism.