  A countably infinite Boolean inverse monoid that can be written as an
increasing union of finite Boolean inverse monoids (suitably embedded) is said
to be of finite type. Borrowing terminology from $C^{\ast}$-algebra theory, we
say that such a Boolean inverse monoid is AF (approximately finite) if the
finite Boolean inverse monoids above are isomorphic to finite direct products
of finite symmetric inverse monoids, and we say that it is UHF (uniformly
hyperfinite) if the finite Boolean inverse monoids are in fact isomorphic to
finite symmetric inverse monoids. We characterize abstractly the Boolean
inverse monoids of finite type and those which are AF and, by using
MV-algebras, we also characterize the UHF monoids.
