Transitive semigroup actions
Charles F. Kelemen · Transactions of the American Mathematical Society · 1969
Following Wallace [15], we define an act to be a continuous function p: Sx X -> X such that (i) S is a topological semigroup, (ii) X is a topological space, and (iii) p(s, p(t, x))=p(st, x) for all s, t e S and xe X.We call (S, X, p) an action triple, X the state space of the act, and we say S acts on X.We assume all spaces are Hausdorff and write sx for p(s, x).S is said to act transitively if Sx = X for all xe X and effectively if sx=tx for all xe X implies that s=t.The first section of this paper deals with transitive actions and especially with the case where the semigroup is simple.We obtain as a corollary that if 5 is a compact connected semigroup acting transitively and effectively on a space A" that contains a cut point, then K, the minimal ideal of S, is a left zero semigroup and X is homeomorphic toK.A C-set is a subset, Y, of X with the property that if M is any continuum contained in X with M n y# 0, then either M<^ Y or F<= M. In the second section, we consider the position of C-sets in the state space and prove as a corollary that if S is a compact connected semigroup with identity acting effectively on the metric indecomposable continuum, X, such that SX= X, then 5 must be a group.