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

Published:2011-01-26
Printed: Jun 2011
• Jonathan David Farley,
Department of Mathematics, California Institute of Technology, Pasadena, California 91125, USA
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## Abstract

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 set
 MSC Classifications: 06D05 - Structure and representation theory 06D50 - Lattices and duality 06A07 - Combinatorics of partially ordered sets