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Computer Science > Discrete Mathematics

arXiv:2109.07189 (cs)
[Submitted on 15 Sep 2021 (v1), last revised 29 Sep 2021 (this version, v2)]

Title:On Characterization of Finite Geometric Distributive Lattices

Authors:Pranab Basu
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Abstract:A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. Kötter and Kschischang proved that codes in the linear lattice can be used for error and erasure-correction in random networks. Codes in the linear lattice have previously been shown to be special cases of codes in modular lattices. Two well known classifications of semimodular lattices are geometric and distributive lattices. Most of the frequently used coding spaces are examples of either or both. We have identified the unique criterion which makes a geometric lattice distributive, thus characterizing all finite geometric distributive lattices. Our characterization helps to prove a conjecture regarding the maximum size of a distributive sublattice of a finite geometric lattice and identify the maximal case. The Whitney numbers of the class of geometric distributive lattices are also calculated. We present a few other applications of this unique characterization to derive certain results regarding linearity and complements in the linear lattice.
Comments: Major corrections included. Abstract and few definitions are modified, some proofs are rectified. A new section added to the previous version. 13 pages, 3 figures, submitted to Journal of Combinatorial Theory, Series A
Subjects: Discrete Mathematics (cs.DM); Information Theory (cs.IT); Combinatorics (math.CO)
MSC classes: 06C10, 06D05
Cite as: arXiv:2109.07189 [cs.DM]
  (or arXiv:2109.07189v2 [cs.DM] for this version)
  https://doi.org/10.48550/arXiv.2109.07189

Submission history

From: Pranab Basu [view email]
[v1] Wed, 15 Sep 2021 10:08:27 UTC (19 KB)
[v2] Wed, 29 Sep 2021 09:02:40 UTC (20 KB)
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