Actions of E-dense semigroups

Abstract We describe the structure of E -dense acts over E -dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We also study cancellative actions of semigroups as a type of generalisation of group actions and characterise locally free cancellative acts over E -dense semigroups that are also E -unitary.