On subalgebras of Boolean interval algebras

We prove that the following three conditions are necessary and sufficient for a Boolean algebra to be embeddable into an interval algebra.

(i)

is generated by a subset such that for all .

(ii)

has a complemented subalgebra lattice, where complements can be chosen in a monotone way.

(iii)

is isomorphic to ClopX for a compact zero-dimensional topological semilattice such that for all .