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 , the semigroup (under composition) of all total transformations of a set , has a proper dense subsemigroup if and only if is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup consisting of all total transformations from into under the operation , where is any fixed element of . We show that this semigroup has a proper dense subsemigroup if and only if and are infinite and , and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.