  We characterize extensions of commutative rings $R \subseteq S$ whose sets of
subextensions $[R,S]$ are finite ({\it i.e.} $R\subseteq S$ has the FIP
property) and are Boolean lattices, that we call Boolean FIP extensions. Some
characterizations involve ``factorial" properties of the poset $[R,S]$. A non
trivial result is that each subextension of a Boolean FIP extension is simple
(i.e. $R \subseteq S$ is a simple pair).
