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Concerning Binary Relations on Connected Ordered Spaces

Published online by Cambridge University Press:  20 November 2018

I. S. Krule
I. S. Krule*
Affiliation:
Louisiana State University
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In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.

In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 107 - 111
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Eilenberg, S., Ordered topological spaces, Amer. J. Math., 63 (1941), 39–45.Google Scholar
2. Faucett, W. M., Compact semigroups irreducibly connected between two idempotents, Proc. Amer. Math. Soc, 6 (1955), 741-7.Google Scholar
3. Krule, I. S., Structs on the 1-sphere, Duke Math. J., 24 (1957), 405–14.Google Scholar
4. Mostert, P. S. and Shields, A. L., On a class of semigroups on En , Proc. Amer. Math. Soc, 7 (1956), 729-34.Google Scholar
5. Wallace, A. D., Struct ideals, Proc. Amer. Math. Soc, 6 (1955), 634-8.Google Scholar