Untranscendable order types

TL;DR

This paper introduces the concept of untranscendability as a multiplicative analogue of additive indecomposability in linear order types, establishing its properties and relationships with indecomposability under various set-theoretic assumptions.

Contribution

It defines untranscendability and $s$-untranscendability, proves their connection to indecomposability, and extends known theorems to these new concepts within set theory.

Findings

Every untranscendable type, except the two-point type, is additively indecomposable.

Every $\sigma$-scattered untranscendable type is strongly indecomposable.

Under the Proper Forcing Axiom, untranscendable Aronszajn types are strongly indecomposable.

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 $\sigma$-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.