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The LMC-compactification of a topological semigroup. (English) Zbl 0655.22001
It is known [J. Berglund, H. Jungheim, and P. Milnes, Compact right topological semigroups and generalizations of almost periodicity (Lect. Notes Math. 663, 1978; Zbl 0406.22005)] that any Hausdorff semitopological semigroup (operation is separately continuous on both sides) has a compactification (e,X) maximal with respect to the property that it is a right topological semigroup (right translations are continuous) and the requirement that $$\lambda_{e(s)}(x)=e(s)x$$ is continuous for each $$s\in S$$. The principal results of this paper show that this conclusion and other analogous ones for other compactifications can be obtained without the assumptions of any continuity or separation.
First, S is assumed to be a semigroup with a topology. It is shown that the compactification (e,X), called the LMC-compactification, for Hausdorff semitopological semigroups is a compactification with respect to the same properties for this S. Second, analogous results are obtained for the almost periodic, strong almost periodic, and weak almost periodic compactifications.
By extending the LMC-compactification to a semigroup with topology, some information about the nature of the mapping e: $$S\to X$$ is lost. Conditions under which e is open or one-to-one are investigated. The paper concludes with some examples that give information on the structure of the LMC-compactification.