Entangledness in Suslin lines and trees

TL;DR

This paper explores the concept of entangledness in Suslin lines and trees, establishing consistency results and generalizations that connect entangled linear orders with properties of $$-trees, and constructing specific Suslin trees with controlled entangledness.

Contribution

It introduces the notion of weakly entangled linear orders, generalizes entangled orders to $$-trees, and constructs Suslin trees with specific entangledness properties.

Findings

Weakly entangled Suslin lines can be consistent.

Entangled $$-trees are characterized as free trees.

Existence of $n$-entangled Suslin trees with special derived 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 $n < \omega$.