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Adjunction semigroups

Published online by Cambridge University Press:  17 April 2009

J. T. Borrego
Affiliation:
University of Massachusetts Amherst, Massachusetts, USA.
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Abstract

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Let S and T be compact (topological) semigroups, A be a closed subsemigroup of S, and f be a continuous homomorphism of A onto T. It is a natural question to ask is there a compact semigroup Z containing T and a continuous homomorphism φ of S onto Z such that φ restricted to A is f?

In the category of compact Hausdorff spaces, the answer to the analogous question may be given by the following construction due to Borsuk. Let X and Y be compact Hausdorff spaces, A be a closed subspace of X and f a continuous function of A onto Y. Then R(f, X) = {(x, y)|f(x) = f(y) or x = y} is a closed equivalence relation on X. The adjunction space Z(f, x) is X/R(f, X). The purpose of this note is to investigate this construction in semigroups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

[1] Borrego, J.T., “Homomorphic retractions in semigroups”, Proc. Amer. Math. Soc. 18 (1967), 716719.CrossRefGoogle Scholar
[2] Borsuk, Karol, “Quelques retracts singulaires”, Fund. Math. 24 (1935), 249258.CrossRefGoogle Scholar
[3] Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups (Math. Surveys 7 (1), Amer. Math. Soc., Providence, 1962).Google Scholar
[4] Paalman-de Miranda, Aida Beatrijs, Topological semigroups (Mathmatisch Centrum, Amsterdam, 1964).Google Scholar
[5] Wallace, A.D., “The structure of topological semigroups”, Bull. Amer. Math. Soc. 61 (1955), 95112.CrossRefGoogle Scholar
[6] Wallace, A.D., “Relative ideals II”, Acta. Math. Acad. Sci. Hungar. 14 (1963), 137148.CrossRefGoogle Scholar