Geodesic
Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order
Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 277-282

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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.
DOI : 10.4153/CMB-2011-002-6
Mots-clés : 06D05, 06D50, 06A07, (distributive) lattice, maximal sublattice, (partially) ordered set 
Farley, Jonathan David. Maximal Sublattices of Finite Distributive Lattices. III: A Conjecture from the 1984 Banff Conference on Graphs and Order. Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 277-282. doi: 10.4153/CMB-2011-002-6
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