Hostname: page-component-54dcc4c588-dbm8p Total loading time: 0 Render date: 2025-09-12T11:44:54.987Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Information

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

1. Baird, B. B., Epimorphisms of inverse semigroups of homeomorphisms between closed subsets, Semigroup Foru. 14 (1977), 161–166.Google Scholar
2. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys of the Amer. Math. Soc. 7 Vol. 2 (Providence, 1967).Google Scholar
3. McAlister, D. B., Homomorphisms of semigroups of binary relations, Semigroup Foru. 3 (1971), 185–188.Google Scholar
4. Thron, W. J., Lattice-equivalence of topological spaces, Duke Mathematical J. 29 (1962), 671–679.Google Scholar