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 $s$-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.