Compact semigroups and suitable sets
Jian He, Karl Heinrich Hofmann, Sally M. Miller, Desmond A. Robbie · TUbilio (Technical University of Darmstadt) · 2002
A suitable set A in a topological semigroup S is a subset of S which contains no idempotents, any limit points of A in S are idempotents, and A, together with all idempotents of S, generates a dense subsemigroup of S. Following work of Hofmann and Morris, who showed that every compact Hausdorff topological group has such a suitable set, this paper extends that result to several classes of compact semigroups all of whose members satisfy S² = S. In particular all compact simple semigroups are shown to have a suitable set. Cartesian products of compact monoids each with a suitable set have suitable sets as do continuous homomorphic images of compact semigroups with suitable sets. It is shown that certain classes of H-chain semigroups have suitable sets. The class of irreducible semigroups falls into two classes, where the members of one class always have a suitable set and in the other class a semigroup which contains no suitable set is constructed. It is shown that compactifications of subsemigroups of Lie groups tend to have suitable sets; these include the `triangle semigroup' as a typical test case. If S is compact, connected, and S² \ e S, then S cannot have a suitable set.