Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-16T05:00:50.452Z Has data issue: false hasContentIssue false
  • English
  • Français

The smallest proper congruence on S(X)

Published online by Cambridge University Press:  18 May 2009

K. H. Hofmann  and
K. D. Magill Jr
K. H. Hofmann
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, Germany (FRG) SUNY at Buffalo, 106 Dlefendorf Hall, Buffalo, New York, 14214-3093, U.S.A.
K. D. Magill Jr
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, Germany (FRG) SUNY at Buffalo, 106 Dlefendorf Hall, Buffalo, New York, 14214-3093, U.S.A.
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.

S(X) is the semigroup of all continuous self maps of the topological space X and for any semigroup S, Cong(S) will denote the complete lattice of congruences on S. Cong(S) has a zero Z and a unit U. Specifically, Z = {(a, a):a ∈ S} and U = S × S. Evidently, Z and U are distinct if S has at least two elements. By a proper congruence on S we mean any congruence which differs from each of these. Since S(X) has more than one element when X is nondegenerate, we will assume without further mention that the spaces we discuss in this paper have more than one point. We observed in [4] that there are a number of topological spaces X such that S(X) has a largest proper congruence, that is, Cong(S(X)) has a unique dual atom which is greater than every other proper congruence on S(X). On the other hand, we also found out in [5] that it is also common for S(X) to fail to have a largest proper congruence. We will see that the situation is quite different at the other end of the spectrum in that it is rather rare for S(X) not to have a smallest proper congruence. In other words, for most spaces X, Cong(S(X)) has a unique atom which is smaller than every other proper congruence.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 3 , September 1988 , pp. 301 - 313
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Aleksandrov, P. S., Combinatorial topology (Graylock Press, 1956).Google Scholar
2.de Groot, J., Groups represented by homeomorphism groups, I, Math. Ann. 138 (1959), 80–102.CrossRefGoogle Scholar
3.Kuratowski, K., Topology, Vol. I (Academic Press, 1966).Google Scholar
4.Magill, K. D. Jr, The largest proper congruence on S(X), Internat. J. Math. Math. Sci. (7) 4 (1984), 663–666.CrossRefGoogle Scholar
5.Magill, K. D. Jr, On a family of ideals of S(X), Semigroup Forum 34 (1987), 321–339.CrossRefGoogle Scholar
6.Thornton, M. C., Semigroups of isotone selfmaps on partially ordered sets, J. London Math. Soc. (2) 14 (1976), 545–553.CrossRefGoogle Scholar