## 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.