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Normal transformation semigroups

Published online by Cambridge University Press:  09 April 2009

J. S. V. Symons
J. S. V. Symons
Affiliation:
University of Western AustraliaNedlands, W. A. 6009.
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Many transformation semigroups S over a set X have the inner-automorphism property: their automorphisms are precisely those mappings of the form g-1 · g: α → g-1αg(α∈S) where g is a fixed permutation of S. For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g-1 · g. We refer to these as -normal transformation semigroups.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 4 , December 1976 , pp. 385 - 390
Copyright
Copyright © Australian Mathematical Society 1976

References

Clifford, A. H. and Preston, G.B. (1967), The algebraic theory of semigroups, II (Math. Surveys of the American Math. Soc. 7, 1967).Google Scholar
Malcev, A. I. (1952), ‘Symmetric groupoids’, Math. Sb., (N.S.) 31, 136–151.Google Scholar
Scott, W. R. (1965), Group Theory (Prentice-Hall, New Jersey, 1965).Google Scholar