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Siblings of Direct Sums of Chains

Davoud Abdi

TL;DR

This work analyzes the number of siblings $Sib(\mathcal{D})$ for direct sums of chains (DSCs) and proves Thomassé's conjecture for countable DSCs by showing $Sib(\mathcal{D})\in\{1,\aleph_0,2^{\aleph_0}\}$. The authors develop a framework based on component-wise embeddings, index-set Cantor–Schröder–Bernstein arguments, and the well-quasi-ordering of countable chains (Laver), to classify when a DSC has finitely many, countably many, or continuum many siblings. They extend the dichotomy to arbitrary DSCs, establishing $Sib(\mathcal{D})\in\{1,\infty\}$ with precise criteria tied to the existence of increasing sequences of nontrivial components, and construct continuum families of non-isomorphic siblings in appropriate cases. The Extensions discuss generalizations to wqo posets and trees, referencing Kruskal’s and Corominas’ results, and pose open questions about direct sums of countable trees and binary relations, indicating rich avenues for further research in equimorphism structures.

Abstract

We prove that a countable direct sum of chains has either one, countably many or else continuum many isomorphism classes of siblings. This proves Thomassé's conjecture for such structures. Further, we show that a direct sum of chains of any cardinality has one or infinitely many siblings, up to isomorphism.

Siblings of Direct Sums of Chains

TL;DR

This work analyzes the number of siblings for direct sums of chains (DSCs) and proves Thomassé's conjecture for countable DSCs by showing . The authors develop a framework based on component-wise embeddings, index-set Cantor–Schröder–Bernstein arguments, and the well-quasi-ordering of countable chains (Laver), to classify when a DSC has finitely many, countably many, or continuum many siblings. They extend the dichotomy to arbitrary DSCs, establishing with precise criteria tied to the existence of increasing sequences of nontrivial components, and construct continuum families of non-isomorphic siblings in appropriate cases. The Extensions discuss generalizations to wqo posets and trees, referencing Kruskal’s and Corominas’ results, and pose open questions about direct sums of countable trees and binary relations, indicating rich avenues for further research in equimorphism structures.

Abstract

We prove that a countable direct sum of chains has either one, countably many or else continuum many isomorphism classes of siblings. This proves Thomassé's conjecture for such structures. Further, we show that a direct sum of chains of any cardinality has one or infinitely many siblings, up to isomorphism.
Paper Structure (6 sections, 22 theorems, 7 equations)

This paper contains 6 sections, 22 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Definitions and Examples
  3. Structural Results
  4. Siblings of Countable DSCs
  5. Siblings of Arbitrary DSCs
  6. Extensions

Key Result

Theorem 1.3

Let C be a chain. Then $Sib(C)=1$ or $\infty$. Moreover, if C is countable, then $Sib(C)=1$, $\aleph_0$ or $2^{\aleph_0}$.

Theorems & Definitions (51)

  • Conjecture 1.1: Thomassé's Conjecture, TH1
  • Conjecture 1.2: The Alternate Thomassé Conjecture
  • Theorem 1.3: LPW
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 41 more