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Untranscendable order types

Garrett Ervin, Alberto Marcone, Thilo Weinert

Abstract

We introduce and study a multiplicative analogue of additive indecomposability for linear order types that we call untranscendability, as well as a strengthening that we call $s$-untranscendability. We show that, with the unique exception of the two-point type, every untranscendable type is additively indecomposable, and every $σ$-scattered untranscendable type is strongly indecomposable. Under the Proper Forcing Axiom, every untranscendable Aronszajn type is strongly indecomposable. We also show that a theorem of Hagendorf and Jullien, that every strictly additively indecomposable type must be strictly indecomposable to either the left or right, has a natural analogue for $s$-untranscendable types.

Untranscendable order types

Abstract

We introduce and study a multiplicative analogue of additive indecomposability for linear order types that we call untranscendability, as well as a strengthening that we call -untranscendability. We show that, with the unique exception of the two-point type, every untranscendable type is additively indecomposable, and every -scattered untranscendable type is strongly indecomposable. Under the Proper Forcing Axiom, every untranscendable Aronszajn type is strongly indecomposable. We also show that a theorem of Hagendorf and Jullien, that every strictly additively indecomposable type must be strictly indecomposable to either the left or right, has a natural analogue for -untranscendable types.
Paper Structure (12 sections, 17 theorems, 1 equation)

This paper contains 12 sections, 17 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic terminology
  4. Sums and products
  5. Ordinal exponentiation and the Cantor normal form
  6. Condensations
  7. Indecomposable types
  8. Basic facts
  9. Strict and non-strict indecomposability
  10. Strong forms of indecomposability
  11. Untranscendable types
  12. Basic facts

Key Result

Proposition 3.2

Suppose that $\varphi$ and $\varphi'$ are equimorphic order types. Then $\varphi$ is indecomposable if and only if $\varphi'$ is indecomposable.

Theorems & Definitions (32)

  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • Corollary 3.7
  • Corollary 3.8
  • ...and 22 more