He, Jian ; Hofmann, Karl H. ; Miller, Sally M. ; Robbie, Desmond A. (2002)
Compact Semigroups and Suitable Sets.
Report, Bibliographie
Kurzbeschreibung (Abstract)
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 $S^2=S$. 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 $\cal 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 $S^2\ne S$, then $S$ cannot have a suitable set.
| Typ des Eintrags: | Report |
|---|---|
| Erschienen: | 2002 |
| Autor(en): | He, Jian ; Hofmann, Karl H. ; Miller, Sally M. ; Robbie, Desmond A. |
| Art des Eintrags: | Bibliographie |
| Titel: | Compact Semigroups and Suitable Sets |
| Sprache: | Englisch |
| Publikationsjahr: | 1 Mai 2002 |
| Ort: | Darmstadt |
| Verlag: | Technische Universität |
| (Heft-)Nummer: | Preprint |
| Reihe: | Preprints Fachbereich Mathematik |
| Band einer Reihe: | 2224 |
| Kollation: | 19 S. |
| Kurzbeschreibung (Abstract): | 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 $S^2=S$. 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 $\cal 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 $S^2\ne S$, then $S$ cannot have a suitable set. |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik |
| Hinterlegungsdatum: | 20 Nov 2008 08:17 |
| Letzte Änderung: | 03 Aug 2023 10:11 |
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