On the grading numbers of direct products of chains

AbstractFor a finite lattice L, denote by l∗(L) and l∗(L) respectively the upper length and lower length of L. The grading number g(L) of L is defined as g(L) = l∗(Sub(L))-l∗(Sub(L)) where Sub(L) is the sublattice-lattice of L. We show that if K is a proper homomorphic image of a distributive lattice L, then l∗(Sub(K)) < l∗(Sub(L)); and derive from this result, formulae for l∗(Sub(L)) and g(L) where L is a product of chains