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The structure of interval orders with no infinite antichain

Maurice Pouzet, Imed Zaguia

Abstract

We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $α$ we exhibit an example of a well-quasi-ordered prime interval order $P_α$ whose chain of maximal antichains has Hausdorff rank $α$.

The structure of interval orders with no infinite antichain

Abstract

We prove that every interval order with no infinite antichain has a Gallai decomposition. That is, is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal we exhibit an example of a well-quasi-ordered prime interval order whose chain of maximal antichains has Hausdorff rank .

Paper Structure

This paper contains 13 sections, 45 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Properties of modules
  4. Basic facts about interval orders
  5. Standard representation of an interval order
  6. A proof of Theorem \ref{['thm:scattered']}
  7. Prime interval orders without infinite antichains are scattered
  8. Prime interval orders without infinite antichains are at most countable
  9. Tree decomposition of graphs with no infinite independent sets
  10. Antichain rank
  11. Proof of Proposition \ref{['prop:finite-chain']}
  12. Gallai decomposition of interval orders without infinite antichains
  13. Examples of well-quasi-ordered prime interval orders whose chain of maximal antichains has arbitrary countable Hausdorff rank

Key Result

Theorem 1

Let $P$ be a poset. Then $P\in IO^{<\omega}$ if and only if $P$ is the lexicographical sum $\sum_{i\in Q} P_{i}$ where each $P_i\in IO^{<\omega}$ and $Q$ is either

Figures (2)

  • Figure 1: The red lines in (b) and (c) indicate linear sum. (a) A well-quasi-ordered prime interval order whose chain of maximal antichains has order type $\omega$. The incomparability graph is an infinite one-way path. (b) A well-quasi-ordered prime interval order whose chain of maximal antichains has order type $\omega+n-1$, for $n\in {\mathbb N}$. (c) A well-quasi-ordered prime interval order whose chain of maximal antichains has order type $\omega\oplus \omega$.
  • Figure 2: The poset $Q$. The $P_j$'s are well-quasi-ordered prime interval orders. The red lines indicate linear sum.

Theorems & Definitions (69)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 59 more