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The smallest proper congruence on S(X)

  • K. H. Hofmann (a1) (a2) and K. D. Magill (a1) (a2)

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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.

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References

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1.Aleksandrov, P. S., Combinatorial topology (Graylock Press, 1956).
2.de Groot, J., Groups represented by homeomorphism groups, I, Math. Ann. 138 (1959), 80102.
3.Kuratowski, K., Topology, Vol. I (Academic Press, 1966).
4.Magill, K. D. Jr, The largest proper congruence on S(X), Internat. J. Math. Math. Sci. (7) 4 (1984), 663666.
5.Magill, K. D. Jr, On a family of ideals of S(X), Semigroup Forum 34 (1987), 321339.
6.Thornton, M. C., Semigroups of isotone selfmaps on partially ordered sets, J. London Math. Soc. (2) 14 (1976), 545553.

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