  Magill proved that the remainders of two locally compact Hausdorff spaces in
their StoneCech compactifications are homeomorphic if and only if the lattices
of their Hausdorff compactifications are lattice isomorphic. His construction
for compactifications are explicitely discussed through the partitions of their
StoneCech compactifications. Partitions in a StoneCech compactification which
lead to Hausdorff compactifications are characterized in this article.
Embeddings of certain upper semi-lattices of compactifications into lattices of
compactifications are constructed.
