Let $(M,\leq,...)$ denote a Boolean ordered o-minimal
structure. We prove that a Boolean subalgebra of $M$ determined by
an algebraically closed subset contains no dense atoms. We show
that Boolean algebras with finitely many atoms do not admit proper
expansions with o-minimal theory. The proof involves decomposition
of any definable set into finitely many pairwise disjoint cells,
i.e., definable sets of an especially simple nature. This leads to
the conclusion that Boolean ordered structures with o-minimal
theories are essentially bidefinable with Boolean algebras with
finitely many atoms, expanded by naming constants. We also discuss
the problem of existence of proper o-minimal expansions of Boolean
algebras.