On the grading numbers of direct products of chains

Abstract

For 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. © 1984.