A Generalisation of the Munn Semigroup

To each meet-semilattice $E$ is associated an inverse semigroup $T_{E}$ called the Munn semigroup of $E$. We generalise this construction by replacing the meet-semilattice $E$ by a presheaf of sets $X$ over a meet-semilattice. The inverse semigroup $T_{X}$ that results is called the generalised Munn semigroup. Our construction can be viewed as a generalisation of one due to Zhitomirskiy as well as a restriction of one due to Reilly. We prove that idempotent-separating representations in to the generalised Munn semigroup characterise \'etale actions of inverse semigroups.