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Point-Transitive Actions by the Unit Interval

  • J. T. Borrego (a1) and E. E. DeVun (a2)

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An action is a continuous function α: T × XX, where T is a semigroup, X is a Hausdorff space, and α(t 1, α(t 2, x)) = α(t 1,t 2 x) for all t 1, t 2T and xX . If, for an action α, Q(α) = {xX| 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].

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References

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1. Aczél, J. and Wallace, A. D., A note on generalizations of transitive systems of transformations, Colloq. Math. 17 (1967), 2934.
2. Cohen, H. and Krule, I. S., Continuous homomorphic images of real clans with zero, Proc. Amer. Math. Soc. 10 (1959), 106109.
3. Day, J. M. and Wallace, A. D., Semigroups acting on continua, J. Austral. Math. Soc. 7 (1967), 327340.
4. Faucett, W. M., Compact semigroups irreducibly connected between two idempotents, Proc. Amer. Math. Soc. 6 (1955), 741747.
5. Hofmann, K. H. and Mostert, P. S., Elements of compact semigroups (Merrill, Columbus, Ohio, 1966).
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), 117143.
7. Paalman-de Miranda, A. B., Topological semigroups (Mathematisch Centrum, Amsterdam, 1964).
8. Stadtlander, D. P., Thread actions, Duke Math. J. 35 (1968), 483490.
9. Wallace, A. D., On the structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955), 95112.
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Point-Transitive Actions by the Unit Interval

  • J. T. Borrego (a1) and E. E. DeVun (a2)

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