  Motivated by some alternatives to the classical logical model of boolean
algebra, this paper deals with algebraic structures which extend skew lattices
by locally invertible elements. Following the meme of the
Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of
isomorphisms) in which the set of objects carries the structure of a skew
lattice. The objects act on the morphisms by left and right restriction and
extension mappings of the morphisms, imitating those of an inductive groupoid.
Conditions are placed on the actions, from which pseudoproducts may be defined.
This gives an algebra of signature (2,2,1), in which each binary operation has
the structure of an orthodox semigroup. In the reverse direction, a groupoid of
the kind described may be reconstructed from the algebra.
