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A universal semigroup

Published online by Cambridge University Press:  17 April 2009

Sidney A. Morris
Sidney A. Morris
Affiliation:
University of Florida, Gainesville, Florida, USA
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Abstract

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J.H. Michael recently proved that there exists a metric semigroup U such that every compact metric semigroup with property P is topologically isomorphic to a subsemigroup of U; where a semigroup S has property P if and only if for each x, y in S, x ≠ y, there is a z in S such that xs ≠ yz or zx ≠ zy

A stronger result is proved here more simply. It is shown that for any set A of metric semigroups there exists a metric semigroup U such that each S in A is topologically isomorphic to a subsemigroup of U. In particular this is the case when A is the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 2 , April 1971 , pp. 159 - 161
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Vol. II (Math. Surveys 7 (II), Amer. Math. Soc., Providence, Rhode Island, 1967).Google Scholar
[2]Kelley, John L.. General topology (Van Nostrand, Toronto, New York, London, 1955).Google Scholar
[3]Michael, J.H., “A universal semigroup”, J. Austral. Math. Soc. 11 (1970), 216–220.CrossRefGoogle Scholar