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Automorphisms of transformation semigroups

Published online by Cambridge University Press:  09 April 2009

R. P. Sullivan
R. P. Sullivan
Affiliation:
University of Western Australia, Nedlands, W. A. 6009
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We let X be an arbitrary non-empty set throughout. Many papers have been written describing the automorphisms of various transformation semigroups defined on X: total (Lyapin (1955), Magill (1967), Malcev (1952), Schreier (1936)), partial (Gluskin (1959), Magill (1967)), partial and 1–1 (Liber (1953)), partial and shifting at most a finite number of elements (Subov (1961a)). In all these cases the automoprhisms are shown to be “inner”, and using the fact this authors deduce that the automorphism group of the given transformation semigroup is isomorphic to the group gx of all permutations defined on X.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 1 , August 1975 , pp. 77 - 84
Copyright
Copyright © Australian Mathematical Society 1975

References

Clifford, A. H. and Preston, G. B. (1961, 1967), The Algebraic Theory of Semigroups, (Math. Surveys, No. 7, Amer. Math. Soc., Providence, RI, Volume, 1 1961: Volume 2, 1967).Google Scholar
Evseev, A. E. and Podran, N. E. (1970), ‘Semigroups of transformations generated by idempotents with given projection characteristics’, Izv. Vyss. Ucebn. Zaved. Mat. 12 (103), 30–36.Google Scholar
Gluskin, L. M. (1959), ‘Ideals of semigroups of transformations’, Mat. Sb. (NS) 47 (89), 111–130.Google Scholar
Howie, J. M. (1966), ‘The subsemigroup generated by the idempotents of a full transformation semigroup’, J. London Math. Soc. 41, 707–716.CrossRefGoogle Scholar
Liber, A. E. (1953), ‘On symmetric generalised groups’, Mat. Sb. (NS) 33 (75), 531–544.Google Scholar
Lyapin, E. S. (1955), ‘Abstract characterisation of certain semigroups of transformations’, Lenningrad Gos. Ped. Inst. Ucen. Zap. 103, 5–29.Google Scholar
Magill, K. D. Jnr, (1966), ‘Automorphisms of the semigroup of all relations on a set’, Canad. Math. Bull. 9, 73–77.CrossRefGoogle Scholar
Magill, K. D. Jnr, (1967), ‘Semigroup Structures for Families of Functions, I. Some Homomorphism Theorems’, J. Austral. Math. Soc. 7, 81–107.CrossRefGoogle Scholar
Malcev, A. I. (1952), ‘Symmetric Groupoids’, Mat. Sb. (NS) 31 (73), 136–151.Google Scholar
Schreier, O. (1936), ‘Über Abbildungen einer abstrakten Menge auf ihre Teilmengen’, Fund. Math. 28, 261–264.CrossRefGoogle Scholar
Scott, W. (1964), Group Theory, (Prentice Hall, 1964).Google Scholar
Sullivan, R. P. (1969), A study in the theory of transformation semigroups, Ph. D. Dissertation, (Monash University August, 1969.)Google Scholar
Sullivan, R. P. (to appear), ‘Automorphisms of relation semigroups’.Google Scholar
Sullivan, R. P. (to appear; a), ‘Ideals in transformation semigroups’.Google Scholar
Sutov, E. G. (1961), ‘Homomorphisms of the semigroup of all partial transformations’, Izeves. Vyss. Ucebn. Zaved. Mat. 3 (22), 177–184.Google Scholar
Sutov, E. G. (1961a), ‘On semigroups of almost identical transformations,’ Dokl. Akad. Nauk. SSSR 134 (1960), 292–295; translated as Soviet Math. Dokl., 1, 1080–1083.Google Scholar