Cancellation in Skew Lattices

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 M3 or N5 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 M3 or N5 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.