  We introduce a new technique for the study of the local extension property
(LEP) for boolean algebras and we use it to show that the clopen algebra of
every compact Hausdorff space $K$ of finite height has LEP. This implies, under
appropriate additional assumptions on $K$ and Martin's Axiom, that every
twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by
Marciszewski and Plebanek.
