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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

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