Reversible and Irreversible Trees

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 $\min\{\alpha ,\lambda\mu\} <\omega$.Comment: 22 page

Splitting families and forcing

AbstractAccording to [M.S. Kurilić, Cohen-stable families of subsets of the integers, J. Symbolic Logic 66 (1) (2001) 257–270], adding a Cohen real destroys a splitting family S on ω if and only if S is isomorphic to a splitting family on the set of rationals, Q, whose elements have nowhere dense boundaries. Consequently, |S|<cov(M) implies the Cohen-indestructibility of S. Using the methods developed in [J. Brendle, S. Yatabe, Forcing indestructibility of MAD families, Ann. Pure Appl. Logic 132 (2–3) (2005) 271–312] the stability of splitting families in several forcing extensions is characterized in a similar way (roughly speaking, destructible families have members with ‘small generalized boundaries’ in the space of the reals). Also, it is proved that a splitting family is preserved by the Sacks (respectively: Miller, Laver) forcing if and only if it is preserved by some forcing which adds a new (respectively: an unbounded, a dominating) real. The corresponding hierarchy of splitting families is investigated