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On the ideal structure of the semigroup of closed subsets of a topological semigroup
Published online by Cambridge University Press: 20 January 2009
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Among the many semigroups which can be derived from a given compact (jointly continuous) semigroup S is the semigroup 2s consisting of its non-empty compact subsets; the product is the usual one defined by the rule EF = {xy:xεE, yεF}. The Vietoris or finite topology on 2s (in which a base for the open sets is obtained by taking all sets of the form 
for l ≦i ≦n} as Vl, V2,…, Vn run over all finite collections of open subsets of S) makes 2s a compact, jointly continuous semigroup. The topology has a long history, having been introduced by Vietoris in 1923 and studied by Michael[4]. The utility of the topological semigroup was established by Hofmann and Mostert [3; see especially Section 3.7]; in fact they prefer to produce directly the uniform structure on 2s rather than the topology.
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 361 - 368
- Copyright
- Copyright © Edinburgh Mathematical Society 1985
References
REFERENCES
1.Hofmann, K. H. and Keimel, K., A general character theory for partially ordered sets and lattices, Memoirs American Math. Soc. 122 (1972).Google Scholar
2.Hofmann, K. H., Mislove, M. and Stralka, A., The Pontryagin duality of compact 0-dimensional semilattices and its applications (Lecture Notes in Mathematics 396, Springer, Berlin, 1974).CrossRefGoogle Scholar
3.Hofmann, K. H. and Mostert, P. S., Elements of compact semigroups (Merrill, Columbus Ohio, 1966).Google Scholar
4.Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182.CrossRefGoogle Scholar
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