Transitive semigroup actions

C. F. Kelemen
1969 Transactions of the American Mathematical Society  
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
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 [16] for semigroups and Stadtlander [12] 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 [2] . 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
doi:10.1090/s0002-9947-1969-0251708-x fatcat:mfhli5d4vjb6fgirrzvqsojpii