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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.

This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains of X meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality N3. Variants of the problem are also considered and open problems specified.

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

Coloring Ordered Sets to Avoid Monochromatic Maximal Chains

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2 results

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

Coloring Ordered Sets to Avoid Monochromatic Maximal Chains