Characterizations of classes of countable Boolean inverse monoids

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.