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

Published online by Cambridge University Press:  20 November 2018

Jonathan David Farley
Jonathan David Farley*
Affiliation:
Institut für Algebra, Johannes Kepler Universität Linz, A-4040 Linz, Österreich, andCenter for International Security and Cooperation, Stanford University, Stanford, California 94305, USAandDepartment of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7, Jamaica, andDepartment of Mathematics, California Institute of Technology, Pasadena, California 91125, USAe-mail: [email protected]
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Abstract

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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.

Keywords

06D0506D5006A07(distributive) latticemaximal sublattice(partially) ordered set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 277 - 282
Copyright
Copyright © Canadian Mathematical Society 2011

References

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