Reversible and Irreversible Trees

Abstract

A tree ${\mathbb T} =\langle T\leq \rangle$ is reversible iff there is no order $\preccurlyeq \;\varsubsetneq \;\leq $ such that ${\mathbb T} \cong \langle T ,\preccurlyeq\rangle$. Using a characterization of reversibility via back and forth systems we detect a wide class of non-reversible trees: ``bad trees" (having all branches of height ${\mathrm{ht}} ({\mathbb T})=|T|=|L_0|$, where $|T|$ is a regular cardinal). Consequently, a countable tree of height $\omega$ and without maximal elements is reversible iff all its nodes are finite. We show that a tree ${\mathbb T}$ is non-reversible iff it contains a ``critical node" or an ``archetypical subtree" (parts of ${\mathbb T}$ with some combinatorial properties). In particular, a tree with finite nodes ${\mathbb T}$ is reversible iff it does not contain archetypical subtrees. Using that characterization we prove that if for each ordinal $\alpha \in [\omega ,{\mathrm{ht}} ({\mathbb T}))$ all nodes of height $\alpha$ are of the same size, or the sequence $\langle \langle |N|,|N\uparrow|\rangle : {\mathcal{N}} ({\mathbb T}) \ni N\subset L_\alpha \rangle $ is finite-to-one, then ${\mathbb T}$ is reversible. Consequently, regular $n$-ary trees are reversible, reversible Aronszajn trees exist and, if there are Suslin or Kurepa trees, there are reversible ones. Also we show that for cardinals $\lambda >1$ and $\mu >0$ and ordinal $\alpha >0$ we have: the tree $\bigcup _\mu {}^{<\alpha }\lambda$ is reversible iff $\min\{\alpha ,\lambda\mu\} <\omega$.