Geodesic
The smallest proper congruence on S(X)
Glasgow mathematical journal, Tome 30 (1988) no. 3, pp. 301-313

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Hofmann, K. H.; Jr, K. D. Magill. The smallest proper congruence on S(X). Glasgow mathematical journal, Tome 30 (1988) no. 3, pp. 301-313. doi: 10.1017/S0017089500007394
@article{10_1017_S0017089500007394,
     author = {Hofmann, K. H. and Jr, K. D. Magill},
     title = {The smallest proper congruence on {S(X)}},
     journal = {Glasgow mathematical journal},
     pages = {301--313},
     year = {1988},
     volume = {30},
     number = {3},
     doi = {10.1017/S0017089500007394},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007394/}
}
TY  - JOUR
AU  - Hofmann, K. H.
AU  - Jr, K. D. Magill
TI  - The smallest proper congruence on S(X)
JO  - Glasgow mathematical journal
PY  - 1988
SP  - 301
EP  - 313
VL  - 30
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007394/
DO  - 10.1017/S0017089500007394
ID  - 10_1017_S0017089500007394
ER  - 
%0 Journal Article
%A Hofmann, K. H.
%A Jr, K. D. Magill
%T The smallest proper congruence on S(X)
%J Glasgow mathematical journal
%D 1988
%P 301-313
%V 30
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007394/
%R 10.1017/S0017089500007394
%F 10_1017_S0017089500007394

[1] 1.Aleksandrov, P. S., Combinatorial topology (Graylock Press, 1956). Google Scholar

[2] 2.de Groot, J., Groups represented by homeomorphism groups, I, Math. Ann. 138 (1959), 80–102. Google Scholar | DOI

[3] 3.Kuratowski, K., Topology, Vol. I (Academic Press, 1966). Google Scholar

[4] 4.Magill, K. D. Jr, The largest proper congruence on S(X), Internat. J. Math. Math. Sci. (7) 4 (1984), 663–666. Google Scholar | DOI

[5] 5.Magill, K. D. Jr, On a family of ideals of S(X), Semigroup Forum 34 (1987), 321–339. Google Scholar | DOI

[6] 6.Thornton, M. C., Semigroups of isotone selfmaps on partially ordered sets, J. London Math. Soc. (2) 14 (1976), 545–553. Google Scholar | DOI

Cité par Sources :