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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

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