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 étale actions of inverse semigroups.