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Point-Transitive Actions by the Unit Interval

Published online by Cambridge University Press:  20 November 2018

J. T. Borrego  and
E. E. DeVun
J. T. Borrego
Affiliation:
University of Massachusetts, Amherst, Massachusetts
E. E. DeVun
Affiliation:
Wichita State University, Wichita, Kansas
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An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t1, α(t2, x)) = α(t1,t2x) for all t1, t2 ∈ T and x ∈ X . If, for an action α, Q(α) = {x ∈ X| α(T × {x}) = X} is non-empty, then α is called a point-transitive action. Our aim in this note is to classify the point-transitive actions of the unit interval with the usual, nil, or min multiplications.

The reader is referred to [5; 7; 9] for information concerning the general theory of semigroups. All semigroups which are considered here are compact and Abelian and all spaces are compact Hausdorff. Actions by semigroups have been studied in [1; 3; 8].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 255 - 259
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Aczél, J. and Wallace, A. D., A note on generalizations of transitive systems of transformations, Colloq. Math. 17 (1967), 29–34.Google Scholar
2. Cohen, H. and Krule, I. S., Continuous homomorphic images of real clans with zero, Proc. Amer. Math. Soc. 10 (1959), 106–109.Google Scholar
3. Day, J. M. and Wallace, A. D., Semigroups acting on continua, J. Austral. Math. Soc. 7 (1967), 327–340.Google Scholar
4. Faucett, W. M., Compact semigroups irreducibly connected between two idempotents, Proc. Amer. Math. Soc. 6 (1955), 741–747.Google Scholar
5. Hofmann, K. H. and Mostert, P. S., Elements of compact semigroups (Merrill, Columbus, Ohio, 1966).Google Scholar
6. Mostert, P. S. and Shields, A. L., On the structure of semigroups on a compact manifold with boundary, Ann. of Math. (2) 65 (1957), 117–143.Google Scholar
7. Paalman-de Miranda, A. B., Topological semigroups (Mathematisch Centrum, Amsterdam, 1964).Google Scholar
8. Stadtlander, D. P., Thread actions, Duke Math. J. 35 (1968), 483–490.Google Scholar
9. Wallace, A. D., On the structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955), 95–112.Google Scholar