  We give a unified treatment of the model theory of various enrichments of
infinite atomic Boolean algebras, with special attention to
quantifier-eliminations, complete axiomatizations and decidability. A classical
example is the enrichment by a predicate for the ideal of finite sets, and a
novel one involves predicates giving congruence conditions on the cardinality
of finite sets. We focus on three examples, and classify them by expressive
power.
