  In this paper we define a monoid called the equivariant Brauer semigroup for
a locally compact Hausdorff groupoid E whose elements consist of Morita
equivalence classes of E-dynamical systems. This construction generalizes both
the equivariant Brauer semigroup for transformation groups and the equivariant
Brauer group for a groupoid. We show that groupoid equivalence induces an
isomorphism of equivariant Brauer semigroups and that this isomorphism
preserves the Morita equivalence classes of the respective crossedproducts,
thus generalizing Raeburn's symmetric imprimitivity theorem.
