  Generalizing de Vries Compactification Theorem and strengthening Leader Local
Compactification Theorem, we describe the partially ordered set $(\LL(X),\le)$
of all (up to equivalence) locally compact Hausdorff extensions of a Tychonoff
space $X$. Using this description, we find the necessary and sufficient
conditions which has to satisfy a map between two Tychonoff spaces in order to
have some kind of extension over arbitrary given in advance Hausdorff local
compactifications of these spaces; we regard the following kinds of extensions:
open, quasi-open, skeletal, perfect, injective, surjective. In this way we
generalize some results of V. Z. Poljakov.
