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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.
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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

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