Abstract. Distributive lattices are well known to be precisely those lattices that possess cancellation: x ∨ y = x ∨ z and x ∧ y = x ∧ z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the 5-element lattices M 3 or N 5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ∧ and ∨ no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M 3 or N 5 that insure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x∨z = y∨z and x∧z = y∧z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.

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We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a noncommutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and socalled override. Our duality shows that there is a canonical choice for this embedding.Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.

In [6] J. Leech introduced skew lattices in rings. In the present paper we study skew lattices in rings of matrices. We prove that every symmetric, normal skew lattice with a finite, distributive maximal lattice image can be embedded in a skew lattice of upper-triangular matrices.