Entangledness in Suslin lines and trees

Abstract

We introduce the idea of a weakly entangled linear order, and show that it is consistent for a Suslin line to be weakly entangled. We generalize the notion of entangled linear orders to ${\omega }_1$-trees, and prove that an ${\omega }_1$-tree is entangled iff it is free. We force the existence of a Suslin tree which is $n$-entangled, but all of whose derived trees of dimension $n+1$ are special, for any positive $n<\omega$.