The present notes are intended as a complement to the existing literature on compact semitopological semigroups and semigroup compactifications; they try to present the major areas of current research on this subject strictly under the point of view of the topological-algebraical strucures involved, avoiding (as much as possible) the use of methods borrowed from functional analysis. The account given is by far not complete, many problems were left open or touched only briefly; the author hopes that this will inspire others to further research.
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