Enrichments of Boolean algebras: a uniform treatment of some classical and some novel examples

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.