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Covering Distributive Lattices by Intervals

Dwight Duffus, Bill Sands

Abstract

We consider the convex subset $[A,B]$ of all elements between two levels $A$ and $B$ of a finite distributive lattice, as a union of (or covered by) intervals $[a,b]$. A 1988 result of Voigt and Wegener shows that for such convex subsets of finite Boolean lattices, covers using $\max(|A|,|B|)$ intervals (the minimum possible number) exist. In 1992 Bouchemakh and Engel pointed out that this result holds more generally for finite products of finite chains. In this paper we show that covers of size $\max(|A|,|B|)$ exist for $[A,B]$ when $A$ is the set of atoms and $B$ the set of coatoms of any finite distributive lattice. This is a consequence of a more general result for finite partially ordered sets. We also speculate on the situation when other levels of finite distributive lattices are considered, and prove a couple of theorems supporting these speculations.

Covering Distributive Lattices by Intervals

Abstract

We consider the convex subset of all elements between two levels and of a finite distributive lattice, as a union of (or covered by) intervals . A 1988 result of Voigt and Wegener shows that for such convex subsets of finite Boolean lattices, covers using intervals (the minimum possible number) exist. In 1992 Bouchemakh and Engel pointed out that this result holds more generally for finite products of finite chains. In this paper we show that covers of size exist for when is the set of atoms and the set of coatoms of any finite distributive lattice. This is a consequence of a more general result for finite partially ordered sets. We also speculate on the situation when other levels of finite distributive lattices are considered, and prove a couple of theorems supporting these speculations.
Paper Structure (5 sections, 10 theorems, 25 equations, 3 figures)

This paper contains 5 sections, 10 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Interval Cover Property and Covers of The Boolean Lattice $\mathcal{B}(n)$
  3. Convex Subsets of Finite Distributive Lattices
  4. Proofs of Theorems \ref{['T:distributive']} and \ref{['T:poset']}
  5. Proofs of Theorems \ref{['T:size2']} and \ref{['T:slight']}

Key Result

Theorem 1

Let ${\mathcal{A}}$ and ${\mathcal{C}}$ be the atoms and coatoms, respectively, of a finite distributive lattice ${\mathcal{D}}$, with $|{\mathcal{D}}| \ge 3$. Then $[{\mathcal{A}},{\mathcal{C}}] = {\mathcal{D}}-\{0,1\}$ is the union of $\max(|{\mathcal{A}}|,|{\mathcal{C}}|)$ intervals.

Figures (3)

  • Figure 1: Covering $[A,C]$ requires $|A| \cdot |C|$ intervals in the lattice $L$
  • Figure 2: Covering $L_2 \cup L_3$ requires 4 intervals. On the right, covering levels $n-1$ and $n$, of sizes $n$ and $m$ respectively, requires $m+n-2$ intervals.
  • Figure 3: Complete the surjection $f:B \mapsto A$ arbitrarily

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 1
  • Theorem 5
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • ...and 8 more