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Dense subsemigroups of generalised transformation semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Amorn Wasanawichit  and
Yupaporn Kemprasit
Amorn Wasanawichit
Affiliation:
Department of Mathematics, Chulalongkorn University, Bangkok 10330, Thailand
Yupaporn Kemprasit
Affiliation:
Department of Mathematics, Chulalongkorn University, Bangkok 10330, Thailand
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Abstract

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In 1986, Higgins proved that T(X), the semigroup (under composition) of all total transformations of a set X, has a proper dense subsemigroup if and only if X is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup T(X, Y) consisting of all total transformations from X into Y under the operation α * β = αθβ, where θ is any fixed element of T(Y, X). We show that this semigroup has a proper dense subsemigroup if and only if X and Y are infinite and | Yθ| = min{|X|,|Y|}, and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 3 , December 2002 , pp. 433 - 446
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math, Surveys 7, Vol. 1 (Amer. Math. Soc., Providence, RI, 1961).Google Scholar
[2]Hall, T. E., ‘Epimorphisms and dominions’, Semigroup Forum 24 (1982), 271–283.CrossRefGoogle Scholar
[3]Higgins, P. M., ‘Dense subsets of some common classes of semigroup’, Semigroup Forum (1986), 5–19.CrossRefGoogle Scholar
[4]Higgins, P. M., Techniques of semigroup theory (Oxford University Press, Oxford, 1992).CrossRefGoogle Scholar
[5]Howie, J. M. and Isbell, J. R., ‘Epimorphisms and dominions II’, J. Algebra 6 (1967), 7–21.CrossRefGoogle Scholar
[6]Isbell, J. R., ‘Epimorphisms and dominions’, in: Proc. Conf. Categorical Algebra (La Jolla, Calif, 1965) (ed. Eilenberg, S. et al. ) (Springer, New York, 1966) pp. 232–246.CrossRefGoogle Scholar
[7]Isbell, J. R., ‘Notes on semigroup dominions’, Semigroup Forum 7 (1974), 364–368.CrossRefGoogle Scholar
[8]Magill, K. D. Jr, ‘Semigroup structures for families of functions, I. Some homomorphism theorems‘, J. Austral. Math. Soc. 7 (1967), 81–94.Google Scholar
[9]Magill, K. D. Jr, ‘Semigroup structures for families of functions, II. Continuous functions’, J. Austral. Math. Soc. 7 (1967), 95–107.Google Scholar
[10]Magill, K. D. Jr, Misra, P. R. and Tewari, U. B., ‘Symons' d-congruence on sandwich semigroups’, Czechoslovak Math. J. 33 (1983), 221–236.CrossRefGoogle Scholar
[11]Sullivan, R. P., ‘Generalised partial transformation semigroups’, J. Austral. Math. Soc. 19 (1975), 470–473.CrossRefGoogle Scholar
[12]Symons, J. S. V., ‘On a generalization of the transformation semigroup’, J. Austral. Math. Soc. (1975), 47–61.CrossRefGoogle Scholar
[13]Vorobev, N. N., ‘On symmetric associative systems’, Leningrad Gos. Ped. Inst. Ucen. Zap. 89 (1953), 161–166 (Russian).Google Scholar