Uncountable Strongly Surjective Linear Orders

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 2 ℵ 0 < 2 ℵ 1 or in the Cohen and other canonical models (where 2 ℵ 0 = 2 ℵ 1 ); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all.