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Piecewise convex embeddability on linear orders

Martina Iannella, Alberto Marcone, Luca Motto Ros, Vadim Weinstein

TL;DR

This work generalizes convex embeddability to the framework of $\mathcal{L}$-convex embeddability, $L \trianglelefteq^{\mathcal{L}} L'$, parametrized by classes $\mathcal{L}$ of linear orders. It identifies the crucial ccs property as precisely the condition for transitivity (and hence being a quasi-order) and develops a comprehensive analysis of the resulting combinatorial structure, including long chains, large antichains, bases, and fractal behavior. The authors explore the descriptive-set-theoretic complexity of these relations, establishing Sigma-1-2 complexity in general and detailing Borel reductions among variants, with sharp contrasts between cases like $\mathcal{L} = \{\mathbf{1}\}$, $\mathcal{L} = \mathsf{Fin}$, and $\mathcal{L} = \mathsf{Lin}$. They also extend the theory to uncountable linear orders, deriving analogous transitivity criteria and showing rich complexity phenomena (including completeness results under certain set-theoretic assumptions) and that there is no finite basis in many uncountable regimes. The paper provides a broad array of concrete ccs examples, including stratifications via $\mathbb{Z}^L$ and Hausdorff ranks, and concludes with open problems motivating further refinement of the ccs criterion and the boundaries of analytic completeness for these generalized embeddability notions.

Abstract

Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$ which are isomorphic to convex subsets of $L'$ ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in arXiv:2309.09910), which are the special cases $\mathcal{L} = \{\mathbf{1}\}$ and $\mathcal{L} = \mathsf{Fin}$. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.

Piecewise convex embeddability on linear orders

TL;DR

This work generalizes convex embeddability to the framework of -convex embeddability, , parametrized by classes of linear orders. It identifies the crucial ccs property as precisely the condition for transitivity (and hence being a quasi-order) and develops a comprehensive analysis of the resulting combinatorial structure, including long chains, large antichains, bases, and fractal behavior. The authors explore the descriptive-set-theoretic complexity of these relations, establishing Sigma-1-2 complexity in general and detailing Borel reductions among variants, with sharp contrasts between cases like , , and . They also extend the theory to uncountable linear orders, deriving analogous transitivity criteria and showing rich complexity phenomena (including completeness results under certain set-theoretic assumptions) and that there is no finite basis in many uncountable regimes. The paper provides a broad array of concrete ccs examples, including stratifications via and Hausdorff ranks, and concludes with open problems motivating further refinement of the ccs criterion and the boundaries of analytic completeness for these generalized embeddability notions.

Abstract

Given a nonempty set of linear orders, we say that the linear order is -convex embeddable into the linear order if it is possible to partition into convex sets indexed by some element of which are isomorphic to convex subsets of ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in arXiv:2309.09910), which are the special cases and . We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.
Paper Structure (9 sections, 55 theorems, 35 equations)

This paper contains 9 sections, 55 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ccs property
  4. Some technical lemmas
  5. Combinatorial properties of $\trianglelefteq^{\mathcal{L}}_\mathsf{LO}$
  6. Borel complexity of $\trianglelefteq^{\mathcal{L}}_\mathsf{LO}$ and $\mathrel{\underline{\bowtie}}^{\mathcal{L}}_\mathsf{LO}$
  7. Examples of ccs families of countable linear orders
  8. Uncountable linear orders
  9. Open problems

Key Result

Proposition 2.5

Theorems & Definitions (120)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6: Sha21
  • Theorem 2.7
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.4
  • ...and 110 more