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Automorphisms of transformation semigroups

Published online by Cambridge University Press:  20 January 2009

S. P. Fitzpatrick  and
J. S. V. Symons
S. P. Fitzpatrick
Affiliation:
University of Western Australia, Nedlands, 6009
J. S. V. Symons
Affiliation:
University of Western Australia, Nedlands, 6009
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It is common property in the theory of transformation semigroups that the presence of all the constant maps ensures that automorphisms are induced by a permutation of the underlying set. Essentially, this goes back to Malcev (2); it has been extensively generalised by Sullivan in (4). For semigroups which do not contain the constants (for example, all surjective transformations of a set, or all injections) there is, as yet, no similar result. The purpose of this note is to provide one.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 4 , September 1975 , pp. 327 - 329
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vols. I and II (American Math. Soc, Providence, Rhode Island, 1961 and 1967).CrossRefGoogle Scholar
(2) Malcev, A. I., Symmetric groupoids, Math. Sb. (N.S.), 31 (73) (1952), 136–151.Google Scholar
(3) Scott, W. R., Group Theory (Prentice-Hall, 1964).Google Scholar
(4) Sullivan, R. P., Automorphisms of Transformation semigroups (to appear).Google Scholar