Uncountable strongly surjective linear orders

Abstract

A linear order $L$ is strongly surjective if $L$ can be mapped onto any of its suborders in an order preserving way. We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, ${♢}^{+}$ implies the existence of a lexicographically ordered Suslin-tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under $2^{{\aleph }_0}<2^{{\aleph }_1}$ or in the Cohen and other canonical models (where $2^{{\aleph }_0}=2^{{\aleph }_1}$); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. We end the paper with a healthy list of open problems.

21 pages, revised version; to appear in Order; comments are very welcome