PreprintThe Work Performed by a Transformation SemigroupJames EastAbstractA partial transformation \al on the finite set {1,\ldots,n} moves an element i of its domain a distance of ii\al units. The \emph{work} w(\al) performed by \al is defined to be the sum of all of these distances. In this article we derive a formula for the total work w(S)=∑_{\al∈ S}w(\al) performed by a subset S of the partial transformation semigroup \PT_{n}. We then obtain explicit formulae for w(S) when S is one of seven important subsemigroups of \PT_{n}: the partial transformation semigroup, the (full) transformation semigroup, the symmetric group, and the symmetric inverse semigroup, as well as their orderpreserving submonoids. Each of these formulae gives rise to a formula for the average work \wb(S)=\frac{1}{S}w(S) performed by an element of S.Keywords: Transformation semigroup, work. AMS Subject Classification: Primary 20M20; Secondary 05A10.
