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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

J. W. Baker
Affiliation:
Department of Pure Mathematics, The University, Sheffield S10 2TN, England
J. S. Pym
Affiliation:
Department of Pure Mathematics, The University, Sheffield S10 2TN, England
H. L. Vaseudeva
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India
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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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

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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), 152182.CrossRefGoogle Scholar