J. Aust. Math. Soc.  73 (2002), 433-445
Dense subsemigroups of generalised transformation semigroups

Amorn Wasanawichit
  Department of Mathematics
  Chulalongkorn University
  Bangkok 10330
  Thailand
 
and
Yupaporn Kemprasit
  Department of Mathematics
  Chulalongkorn University
  Bangkok 10330
  Thailand
 


Abstract
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 $\alpha * \beta=\alpha\theta\beta$, where $\theta$ 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 $|Y\theta|=\min\{|X|, |Y|\}$, and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.
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