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Generalised partial transformation semigroups

Published online by Cambridge University Press:  09 April 2009

R. P. Sullivan
R. P. Sullivan
Affiliation:
University of Papua & New GuineaBoroko T.P.N.G.
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It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 4 , June 1975 , pp. 470 - 473
Copyright
Copyright © Australian Mathematical Society 1975

References

Clifford, A. H. and Preston, G. B. (1961), The Algebraic Theory of semigroups, Vol. 1 (Math. Surveys of the Amer. Math. Soc., 1961).Google Scholar
Clifford, A. H. and Preston, G. B. (1967), The Algebraic Theory of Semigroups, Vol. 2 (Math. Surveys of the Amer. Math. Soc. 1967).Google Scholar
Ljapin, E. S., Semigroups (1963), Vol. 3 (Translations of Math. Monographs, Amer. Math. Soc., 1963).Google Scholar
Magill, K. D. Jr (1967), ‘Semigroup structures for families of functions, 1, Some homomorphism theorems’, J. Austral. Math. Soc. 7, 81–94.Google Scholar