I take the points of the Stone-Cech Compactification of a semigroup T to be the set of all ultrafilters on T, the principal ultrafilters being identified with the points of T. There is a natural extension of the operation of T to its Stone- Cech Compactification, making it a compact right topological semigroup. A subset V of a semigroup T is called a left ideal if it is nonempty and TV is a subset of V. It is called a right ideal if it is nonempty and VT is a subset of V. The intersection of any minimal left ideal and minimal right ideal is a group, called a maximal group. However, the structure of these groups is not guaranteed to be an interesting one. For some semigroups, this group may be relatively large and for others it may be small. An important class of semigroups is the free semigroups. When A is a nonempty set, the free semigroup on the alphabet A is the set of all words on A. In this book, it is shown that it is consistent that there exists a special type of ultrafilter for which the maximal group is a single element.

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