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The partially ordered family of all ℛ-classes of S(x)

Published online by Cambridge University Press:  20 January 2009

K. D. Magill Jr
K. D. Magill Jr
Affiliation:
Suny at Buffalo
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Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the following

Conjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:

(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.

(2) ℛ(S(X)) and (S{Y)) are order isomorphic.

(3) The semigroupsS(X) and S(Y) are isomorphic.

(4) The spaces X and Y are homeomorphic.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 1 , February 1983 , pp. 7 - 13
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

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