On subalgebras of Boolean interval algebras

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

(i)

$A$ is generated by a subset $R$ such that $r\cdot s \in \{0,r,s\}$ for all $r,s\in R$.

(ii)

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

(iii)

$A$ is isomorphic to ClopX for a compact zero-dimensional topological semilattice $(X; \cdot )$ such that $x\cdot y\cdot z \in \{x\cdot y, x\cdot z\}$ for all $x,y,z \in X$.