Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation: x úy = x úzx \lor y = x \lor z and x ùy = x ùzx \land y = x \land z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-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 ù\land and ú\lor 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 ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x úz = y úzx \lor z = y \lor z and x ùz = y ùzx \land z = y \land 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.