Cancellation in skew lattices (Q633218)

A skew lattice is an algebra \((L;\vee, \wedge)\) of type \((2,2)\) such that both operations are associative, idempotent and satisfy the absorbtion identities \(x\wedge(x\vee y)=x=(y\vee x)\wedge x\) and \(x\vee(x\wedge y)=x=(y\wedge x)\vee x\). Clearly, a lattice is a skew lattice. A skew lattice is called left cancellative whenever \(x\vee y=x\vee z\) and \(x\wedge y=x\wedge z\) imply \(y=z.\) Similarly, we can define right cancellative skew lattices and, eventually, (fully) cancellative skew lattices. The aim of this paper is to study and characterize various forms of cancellation. Here are two typical results: (1) Left (right, fully) cancellative skew lattices form a variety. (2) Let \(L\) be a skew lattice. Then \(L\) is left (right, fully) cancellative if and only if \(L\) does not contain as subalgebras \(M_3\) (= diamond), \(N_5\) (= pentagon) and four additional special finite skew lattices.