On monothetic semigroups
Robert J. Koch · Proceedings of the American Mathematical Society · 1957
By semigroup we shall mean a Hausdorff space together with a continuous associative multiplication. The study of monothetic semigroups has been initiated independently by several authors; most of the known results involve some form of compactness. We repeat here some of these known results for the sake of completeness. Among the results we establish is the fact that a compact monothetic semigroup with two distinct generators is a group. Some results are given on translates of compact sets in semigroups. Finally we establish analogs for semigroups of Weil's theorem on monothetic groups. We use S throughout to denote a semigroup, and follow the terminology of Clifford [1]. In particular a unit is a two-sided identity element and K is the kernel ( =minimal ideal) of S (if it is not empty). For aES we let r(a) denote the closure of the set of positive powers of a. We say S is monothetic if for some aES, r(a) =S. In such a case we say that a is a generator of S. Note that if S is monothetic then it is commutative. In the group case it is customary to use both positive and negative powers to define monotheticity [5 ]. It can be seen that the two notions agree in a compact group. To eliminate confusion we may for emphasis write monothetic (+) or monothetic (?). For the terminology of nets we follow Kelley [6]. Compactness is used in the sense of bicompactness. It is a pleasure to record our obligations to Professor A. D. Wallace. The following basic result on monothetic semigroups is due to Numakura [2], Peck [10], and the author independently; there is also a generalization due to Wallace [12 ].