The class of Aronszajn lines under epimorphisms
Lucas Polymeris, Carlos Martinez-Ranero
TL;DR
The paper investigates the epimorphism-based order $\trianglelefteq$ on uncountable linear orders, focusing on Aronszajn and Countryman lines and contrasting it with embeddability $\preceq$. It shows that under $\mathsf{MA}_{\aleph_{1}}$ there exists a strongly surjective Countryman line, while the $\trianglelefteq$-structure of Aronszajn lines is not well-quasi-ordered in ZFC (infinite antichains) and can exhibit infinite decreasing chains under $\mathsf{MA}_{\aleph_{1}}$; under $\mathsf{PFA}$ a two-element $\trianglelefteq$-basis for Aronszajn lines is obtained via a Countryman line $C$ and its dual $C^{\star}$, with a broader analogy to the countable case. The authors develop a robust decomposition framework to realize prescribed endpoint configurations and apply forcing (Moore's forcing and epimorphism-adding forcings) to produce epimorphisms and chains, yielding both positive results and obstructions. The work culminates in results about universality and basis questions, and ends with open problems on strong surjectivity of universal Aronszajn lines under $\mathsf{PFA}$ and the existence of irreversible Aronszajn lines.
Abstract
A linear order $A$ is called strongly surjective if for every non empty suborder $B \preceq A$, there is an epimorphism from $A$ onto $B$ (denoted by $B \trianglelefteq A$). We show, answering some questions of Dániel T. Soukup, that under $\mathsf{MA}_{\aleph_{1}}$ there is a strongly surjective Countryman line. We also study the general structure of the class of Aronszajn lines under $\trianglelefteq$, and compare it with the well known embeddability relation $\preceq$. Under $\mathsf{PFA}$, the class of Aronszajn lines and the class of countable linear orders enjoy similar nice properties when viewed under the embeddability relation; both are well-quasi-ordered and have a finite basis. We show that this analogy does not extend perfectly to the $\trianglelefteq$ relation; while it is known that the countable linear orders are still well-quasi-ordered under $\trianglelefteq$, we show that already in $\mathsf{ZFC}$ the class of Aronszajn lines has an infinite antichain, and under $\mathsf{MA}_{\aleph_{1}}$ an infinite decreasing chain as well. We show that some of the analogy survives by proving that under $\mathsf{PFA}$, for some carefully constructed Countryman line $C$, $C$ and $C^{\star}$ form a $\trianglelefteq$-basis for the class of Aronszajn lines. Finally we show that this does not extend to all uncountable linear orders by proving that there is never a finite $\trianglelefteq$-basis for the uncountable real orders.