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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

REFERENCES

1.Borsuk, K., Theory of Retracts (Polska Akademia Nauk, Monografie Matematyczne, PolishScientific Publishers, Warszawa, 1967).Google Scholar
2.Gillman, L. and Jerison, M., Rings of Continuous Functions (D. Van Nostrand, New York, 1960).CrossRefGoogle Scholar
3.Hofmann, K. H. and Mostert, P. S., Elements of Compact Semigroups (Charles E. Merrill Books. Inc., Columbus, Ohio, 1966).Google Scholar
4.Kuratowski, K., Topology, Vol. I (Academic Press, New York, 1966).Google Scholar
5.Magill, K. D. JR., Greens ℒ-relation for semigroups and compactifications, Aeq. Math. 19 (1979), 123–133.CrossRefGoogle Scholar
6.Magill, K. D. JR., Some Open Problems and Directions for Further Research in Semigroups of Continuous Selfmaps, Universal Alg. and App., Banach Cent. Pub., Vol. 9, PWN-Polish Sci. Pub., Warsaw (1980), 439–453.Google Scholar
7.Magill, K. D., and Subbiah, S., Green's relation for regular elements of semigroups ofendomorphisms, Can. J. Math. 26 (1974), 1484–1497.CrossRefGoogle Scholar
8.Magill, K. D. JR., and Subbiah, S., The partially ordered family of regular ℛ-classes of S(X), Aeq. Math. 15 (1977), 91–97.CrossRefGoogle Scholar
9.Weir, M. D., Hewitt-Nachbin Spaces (North-Holland/American Elsevier, Inc., New York, 1975).Google Scholar
10.Whyburn, G. T., Analytic Topology (Colloq. Pub. Vol. 28, Prov., R.I., Amer. Math. Soc., 1963).Google Scholar