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Inverse Semigroups of Homeomorphisms are Hopfian

Published online by Cambridge University Press:  20 November 2018

Bridget B. Baird
Bridget B. Baird*
Affiliation:
University of Florida, Gainesville, Florida
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If X is a nonempty topological T1 space then the set of all homeomorphisms whose domains and ranges are closed subsets of X forms a semigroup under partial composition of functions. We call it IF(X). If, in a semigroup, every element a is matched with a unique element b such that aba = a and bab = b then the semigroup is an inverse semigroup (b is called the inverse of a and is denoted by a−1). We have that IF(X) is an inverse semigroup with the algebraic inverse of a map ƒ being just the inverse map ƒ-1. In this paper we examine epimorphisms from IF(X) onto IF(Y). The main theorem gives conditions under which an epimorphism must be an isomorphism.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 800 - 807
Copyright
Copyright © Canadian Mathematical Society 1979

References

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