Enrichments of Boolean algebras by Presburger predicates

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. Our main enrichment is by a predicate for the ideal of finite sets and predicates for congruence conditions on the cardinalities of finite sets, but we also give new proofs of some classical results. We then classify and compare the expressive power of the enriched theories.