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Chapter 3 - Lattices

Published online by Cambridge University Press:  05 May 2010

Neil White
Affiliation:
University of Florida
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Summary

One of the ways to describe a matroid M(E) in Chapter 2 was by means of a closure operator cl on the set of subsets of E. This closure operator distinguishes a closed set or flat of the matroid M(E) as a set TE with the property T = cl(T). In this chapter we want to study the collection L(M) of flats of M(E) and find out how much of the structure of M(E) is reflected in the structure of L(M).

L(M) is (partially) ordered by set-theoretic inclusion. Furthermore, there are two natural binary operations on L(M): For every pair T1, T2 of flats of M(E) define T1T2 as the smallest flat containing both T1 and T2, and T1T2 as the largest flat contained in both T1 and T2. [It is not difficult to see that these operations are well defined. In fact, T1T2 = cl(T1T2), and T1T2 = T1T2.]

The appropriate setting for study of this algebraic structure on L(M) is lattice theory. So we shall briefly introduce the concept of a lattice and then proceed to investigate special classes of lattices. The guiding example thereby is the lattice of subspaces of a finite-dimensional vector space. Such a lattice is, in particular, modular. Hence, we shall look at the concept of modularity in a lattice, with special emphasis on the notion of a modular pair of elements.

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Theory of Matroids , pp. 45 - 61
Publisher: Cambridge University Press
Print publication year: 1986

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  • Lattices
  • Edited by Neil White, University of Florida
  • Book: Theory of Matroids
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629563.006
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  • Lattices
  • Edited by Neil White, University of Florida
  • Book: Theory of Matroids
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629563.006
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Lattices
  • Edited by Neil White, University of Florida
  • Book: Theory of Matroids
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511629563.006
Available formats
×