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1. CMB 2011 (vol 54 pp. 277)

Farley, Jonathan David
 Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order 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 setCategories:06D05, 06D50, 06A07
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