Cancellation in Skew Lattices

Abstract

Distributive lattices are well known to be precisely those lattices that possess cancellation: $x \lor y = x \lor z$ and $x \land y = x \land z$ 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 $\land$ and $\lor$ 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 ($x \lor z = y \lor z$ and $x \land z = y \land z$ 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.