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

http://dx.doi.org/10.4153/CMB-2011-002-6
Canad. Math. Bull. 54(2011), 277-282

  Published:2011-01-26
 Printed: Jun 2011

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.