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Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order
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- Journal:
- Canadian Mathematical Bulletin / Volume 54 / Issue 2 / 01 June 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 277-282
- Print publication:
- 01 June 2011
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- Article
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Let
$L$ be a finite distributive lattice. Let 
$\text{Su}{{\text{b}}_{0}}(L)$ be the lattice
$$\{S\,|\,S\,\text{is}\,\text{a sublattice of }L\}\cup \{\phi \}$$and let
${{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(L)]$ be the length of the shortest maximal chain in $\text{Su}{{\text{b}}_{0}}(L)$. It is proved that if 
$K$ and $L$ are non-trivial finite distributive lattices, then
$${{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(K\times L)]={{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(K)]+{{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(L)]$$A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.
