LAUSR

We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f(x) ≤ f(y). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. 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 2ℵ0<2ℵ1$2^{\aleph _{0}}<2^{\aleph _{1}}$ or in the Cohen and other canonical models (where 2ℵ0=2ℵ1$2^{\aleph _{0}}= 2^{\aleph _{1}}$); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.