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Monomorphisms of Semigroups of Local Dendrites

Published online by Cambridge University Press:  20 November 2018

K. D. Magill JR.
K. D. Magill JR.*
Affiliation:
State University of New York at Buffalo, Buffalo, New York
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When we speak of the semigroup of a topological space X, we mean S(X) the semigroup of all continuous self maps of X. Let h be a homeomorphism from a topological space X onto a topological space Y. It is immediate that the mapping which sends f ∊ S(X) into h º f º h−1 is an isomorphism from the semigroup of X onto the semigroup of Y. More generally, let h be a continuous function from X into Y and k a continuous function from Y into X such that k º h is the identity map on X. One easily verifies that the mapping which sends f into h º f º k is a monomorphism from S(X) into S(Y). Now for “most” spaces X and Y, every isomorphism from S(X) onto S(Y) is induced by a homeomorphism from X onto Y. Indeed, a number of the early papers dealing with S(X) were devoted to establishing this fact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 4 , 01 August 1986 , pp. 769 - 780
Copyright
Copyright © Canadian Mathematical Society 1986

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