Local extension property for finite height spaces

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.