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Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings

V. Ironmonger, N. Ruskuc

TL;DR

The paper establishes decidability results for two classical order-theoretic properties—well quasi-order (wqo) and atomicity—for downward-closed sets of finite equivalence relations under both non-consecutive and consecutive embedding orders. It develops a unified framework based on factor graphs and path posets to translate questions about Av-sets into graph-theoretic criteria (e.g., absence of in–out cycles, presence of ambivalent vertices, and strong connectivity or bicycles). A complete characterization and decidability results are obtained: (i) wqo for the non-consecutive order and an atomicity criterion tied to uniformity of forbidden-class sizes, (ii) decidable wqo and atomicity for the consecutive order, via handling unambiguous and coloured variants of the factor graphs. The approach generalizes Higman-type arguments to structured finite-relations settings and provides a solid basis for open problems and extensions to other relational structures and embedding notions.

Abstract

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations $ρ_1,\dots,ρ_k$, is the downward closed set Av$(ρ_1,\dots,ρ_k)$ consisting of all equivalence relations which do not contain any of $ρ_1,\dots,ρ_k$: (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?

Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings

TL;DR

The paper establishes decidability results for two classical order-theoretic properties—well quasi-order (wqo) and atomicity—for downward-closed sets of finite equivalence relations under both non-consecutive and consecutive embedding orders. It develops a unified framework based on factor graphs and path posets to translate questions about Av-sets into graph-theoretic criteria (e.g., absence of in–out cycles, presence of ambivalent vertices, and strong connectivity or bicycles). A complete characterization and decidability results are obtained: (i) wqo for the non-consecutive order and an atomicity criterion tied to uniformity of forbidden-class sizes, (ii) decidable wqo and atomicity for the consecutive order, via handling unambiguous and coloured variants of the factor graphs. The approach generalizes Higman-type arguments to structured finite-relations settings and provides a solid basis for open problems and extensions to other relational structures and embedding notions.

Abstract

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations , is the downward closed set Av consisting of all equivalence relations which do not contain any of : (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?
Paper Structure (11 sections, 36 theorems, 26 equations, 6 figures)

This paper contains 11 sections, 36 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. WQO under the non-consecutive embedding ordering
  4. Atomicity under the non-consecutive embedding ordering
  5. Digraphs: definitions and some useful results
  6. The Factor Graph of an Avoidance Set
  7. Two types of cycles which imply non-wqo
  8. Coloured Equivalence Relations
  9. WQO under the Consecutive Embedding Order
  10. Atomicity under the consecutive embedding ordering
  11. Concluding remarks and open problems

Key Result

Proposition 2.9

A downward closed set $C$ in a poset $(X, \leq)$ is atomic if and only if $C$ satisfies the joint embedding property (JEP): for every pair of elements $x, y \in C$ there is an element $z \in C$ such that $x\leq z$ and $y \leq z$.

Figures (6)

  • Figure 1: The factor graph of $\mathop{\mathrm{Av}}\nolimits(\lvert \space1 \space 2\space \lvert \space 3 \space \lvert)$.
  • Figure 2: The factor graph of $\mathop{\mathrm{Av}}\nolimits(B)$ from Example \ref{['ex factor graph 2']}.
  • Figure 3: The factor graph of $\mathop{\mathrm{Av}}\nolimits(B)$ from Example \ref{['ex active classes']}.
  • Figure 4: The uncoloured and coloured factor graphs of $\mathop{\mathrm{Av}}\nolimits(B)$ from Example \ref{['ex col graph']}.
  • Figure 5: The factor graph of $\mathop{\mathrm{Av}}\nolimits(\lvert \space 1 \space 2 \space3\space \lvert, \lvert \space 1 \space 3\space \lvert \space 2 \space \lvert)$.
  • ...and 1 more figures

Theorems & Definitions (111)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9
  • Example 2.10
  • ...and 101 more