A nonassociative extension of the class of distributive lattices

Let Z = {O, 1, 2} and define two binary operations /\ and V on Z as follows: 0/\ 1 = 0,0 V 1 = 1, 1 /\ 2 = 1,1 V 2 = 2,2 A 0 = 2, 2 V 0 = 2, both operations are idempotent and commutative. This paper deals with the equational class Z generated by the algebra . The class Z contains the class of all distributive lattices and Z is a subclass of the class of weakly associative lattices (trellis, T·lattice) in the sense of E. Fried and H. Skala. The purpose of this paper is to prove that Z shares the most important properties of the class of distributive lattices.