Cancellation in Skew Lattices

摘要

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 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 A and v 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∨ 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.