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The semigroup of continuous selfmaps of I has infinitely many ideals

Published online by Cambridge University Press:  18 May 2009

J. W. Baker  and
J. S. Pym
J. W. Baker
Affiliation:
Department of Pure Mathmatics, The University, Sheffield S3 7RH, England
J. S. Pym
Affiliation:
Department of Pure Mathmatics, The University, Sheffield S3 7RH, England
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Let I denote the interval [0,1] in its usual topology and let S = S(I) be the semigroup of continuous mappings of I into itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties of S are still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing that S has infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5: S has infinitely many distinct congruences. We believe that S must have c distinct ideals, but we have been unable to prove this.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 27 - 33
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Forouzanfar, A. M., Closed ideals in semigroups of continuous selfmaps, Semigroup Forum, to appear.Google Scholar
2.Hu, S.-T., Homotopy theory (Academic Press, 1959).Google Scholar
3.Magill, K. D. Jr, A survey of semigroups of continuous selfmaps, Semigroup Forum 11 (1975/1976), 189–282.Google Scholar
4.Magill, K. D. Jr, On the ideals of the semigroup of the l-sphere, Czech. Math. J. 39 (114) 1989, 248–261.Google Scholar
5.Magill, K. D. Jr, Congruences on S(X), in The analytical and topological theory of semigroups—Trends and developments, ed. Hofmann, K. H. et al. (de Gruyter, Berlin, 1990), pp. 379–396.CrossRefGoogle Scholar