Abstract
AbstractIn 1986, Higgins proved thatT(X), the semigroup (under composition) of all total transformations of a setX, has a proper dense subsemigroup if and only ifXis infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroupT(X, Y) consisting of all total transformations fromXintoYunder the operation α * β = αθβ, where θ is any fixed element ofT(Y, X). We show that this semigroup has a proper dense subsemigroup if and only ifXandYare 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.
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