On the grading numbers of direct products of chains (Q793064)

A finite lattice \({\mathcal L}\) is considered. The concepts of upper length and lower length of \({\mathcal L}\) are introduced respectively as the length of a longest and shortest maximal chain in the sublattice-lattice of \({\mathcal L}\). The grading number of \({\mathcal L}\) is defined as the difference between the upper and lower lengths of \({\mathcal L}\). It is shown that if \({\mathcal K}\) is a proper homomorphic image of a distributive lattice \({\mathcal L}\), then the lower length of \({\mathcal K}\) is less than the lower length of \({\mathcal L}\). From this result a formula for lower length and grading number of \({\mathcal L}\) when \({\mathcal L}\) is a direct product of chains is obtained. Corresponding problems are set up.