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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 D\'aniel 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 $C$, $C$ an

Author

Polymeris, Lucas; Martinez-Ranero, Carlos

Published

2025

Language

EN