Compactifications and generalized essential semilattice homomorphisms

A method for constructing compactifications of a locally compact space X with a given remainder Y has been recently introduced. Given a map satisfying some properties (Essential Semilattice Homorphism) from a basis of Y to the set of non-relatively compact open subsets of X, one can put a suitable topology on the union of X and Y and obtain a compactification of X (ESH-compactification). If we are given a map p with the same properties but such that the image also contains relatively compact open sets, we can obtain a compactification of X whose remainders is a subset of Y, the so-called singular set. In this paper this kind of compactifications (GESH-compactifications) are introduced and studied. We give some conditions for a GESH-compactification being an ESH-compactification. We deduce necessary and sufficient conditions for a compactification to be an ESH-compactification. Finally we prove that the supremum of a family F of ESH-compactifications is a GESH-compactification induced by a map defined by mean of the maps that induce the compactifications of the family F.