On the grading numbers of direct products of chains
type: Item
instance of: scholarly article
title: On the grading numbers of direct products of chains
zbMATH Open document ID: 0538.06007
DOI: 10.1016/0012-365X(84)90147-X
author: N. E. Zubov
published in: Discrete Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 3855176
zbMATH Keywords: upper length
zbMATH Keywords: lower length
zbMATH Keywords: maximal chain
zbMATH Keywords: sublattice-lattice
zbMATH Keywords: grading number
zbMATH Keywords: direct product of chains
reviewed by: Frada Burstein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3855176