Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T08:12:46.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotone Homomorphisms of Compact Semigroups

Published online by Cambridge University Press:  09 April 2009

C. E. Clark
C. E. Clark
Affiliation:
University of Missouri
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.

The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 1-2 , February 1969 , pp. 167 - 175
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Anderson, L. W., and Hunter, R. P., ‘Homomorphisms and Dimension’, Math. Annalen. 167 (1962), 248–268.CrossRefGoogle Scholar
[2]Brown, D. R., ‘Topological Semilattices on the Two-cell’, Pacific Journal of Math. 15 (1965), 35–46.CrossRefGoogle Scholar
[3]Clifford, A. H., ‘Naturally Totally Ordered Commutative Semigroups’, Am. J. of Math. 56 (1954), 631–646.CrossRefGoogle Scholar
[4]Cohen, H., and Krule, I. S., ‘Continuous Homomorphic Images of Real Clans with Zero’, Proc. Am. Math. Soc. 10 (1959), 106–109.CrossRefGoogle Scholar
[5]Koch, R. J., ‘On Monothetic Semigroups’, Proc. Am. Math. Soc. 10 (1959), 648–655.Google Scholar
[6]Mostert, P. S., and Shields, A. L., ‘On the Structure of Semigroups on a Compact Manifold with Boundary’, Annals of Math. 64 (1957), 117–143.CrossRefGoogle Scholar
[7]Phillips, R. C., ‘Interval Clans with Non-degenerate Kernel’, Proc. Am. Math. Soc. 14 (1963), 396–400.CrossRefGoogle Scholar
[8]Wallace, A. D., ‘Factoring a Lattice’, Proc. Am. Math. Soc. 9 (1958), 250–252.CrossRefGoogle Scholar
[9]Wallace, A. D., ‘Two Theorems on Topological Lattices’, Pacific Journal of Math. 7 (1957), 1239–1241.CrossRefGoogle Scholar
[10]Whyburn, G. T., Analytic Topology (Providence, Rhode Island, American Math. Soc. Colloquium Publications, Vol. 28, 1942).Google Scholar