### Transitive semigroup actions

C. F. Kelemen
1969 Transactions of the American Mathematical Society
Following Wallace , 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
more » ... is 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. The author wishes to thank Professor L. W. Anderson for his patient advice and criticism. Definitions and notation. The notation is generally that of Wallace  for semigroups and Stadtlander  for actions. Let S be a topological semigroup then we denote by K(S) the unique minimal ideal (if it exists) of S and by E(S) the set of idempotents of 5. When the semigroup referred to is clear, the above will be shortened to K and E respectively. We recall that if S is compact then K(S) exists and is closed and F(S)# 0. For each e e E(S), H(e) denotes the maximal subgroup of S containing e. S is a left zero semigroup if xy=x for all x, y e S. A left group is a semigroup that is left simple and right cancellative; it is isomorphic to Ex G where £ is a left zero semigroup, G is a group and multiplication is coordinate wise  . An algebraic isomorphism that is simultaneously a topological homeomorphism is called an iseomorphism. The Q-set of the action triple (S, X, p) is the set Q={x e X | Sx=X), thus if Q = X the action is transitive. The action triple (S, X, p) is said to be equivalent to