CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order

  Published:2011-01-26
 Printed: Jun 2011
  • Jonathan David Farley,
    Department of Mathematics, California Institute of Technology, Pasadena, California 91125, USA
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

Let $L$ be a finite distributive lattice. Let $\operatorname{Sub}_0(L)$ be the lattice $$ \{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\} $$ and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are non-trivial finite distributive lattices, then $$ \ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)]. $$ A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.
Keywords: (distributive) lattice, maximal sublattice, (partially) ordered set (distributive) lattice, maximal sublattice, (partially) ordered set
MSC Classifications: 06D05, 06D50, 06A07 show english descriptions Structure and representation theory
Lattices and duality
Combinatorics of partially ordered sets
06D05 - Structure and representation theory
06D50 - Lattices and duality
06A07 - Combinatorics of partially ordered sets
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/