The smallest ideal K(βS) of the Stone-Čech compactification of a discrete semigroup S is the union of pairwise isomorphic and homeomorphic minimal left ideals. We provide characterizations of semigroups S that have the property that the minimal left ideals of βS have isolated points, provide details about the structure of K(βS) for such semgiroups, and in some instances provide explicit descriptions of K(βS).

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