Reversible Disjoint Unions of Well Orders and Their Inverses

Abstract

A poset ${\mathbb{P}}$ is called reversible iff every bijective homomorphism $f:{\mathbb{P}} \rightarrow {\mathbb{P}}$ is an automorphism. Let ${\mathcal{W}}$ and ${\mathcal{W}} ^*$ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form ${\mathbb{P}} =\bigcup _{i\in I}{\mathbb{L}} _i$, where ${\mathbb{L}} _i, i\in I$, are pairwise disjoint linear orders from ${\mathcal{W}} \cup {\mathcal{W}} ^*$. First, if ${\mathbb{L}} _i \in {\mathcal{W}}$, for all $i\in I$, and ${\mathbb{L}} _i \cong \alpha _i =\gamma_i+n_i\in Ord$, where $\gamma_i\in Lim \cup \{0\}$ and $n_i\in\omega$, defining $I_\alpha := \{ i\in I : \alpha_i = \alpha \}$, for $\alpha \in Ord$, and $J_\gamma := \{ j\in I : \gamma_j = \gamma \}$, for $\gamma\in Lim _0$, we prove that $\bigcup _{i\in I} {\mathbb{L}} _i$ is a reversible poset iff $\langle \alpha _i :i\in I\rangle$ is a finite-to-one sequence, or there is $\gamma =\max \{ \gamma _i : i\in I\}$, for $\alpha \leq \gamma$ we have $|I_\alpha |<\omega $, and $\langle n_i : i\in J_\gamma \setminus I_\gamma \rangle $ is a reversible sequence of natural numbers. The same holds when ${\mathbb{L}} _i \in {\mathcal{W}} ^*$, for all $i\in I$. In the general case, the reversibility of the whole union is equivalent to the reversibility of the union of components from ${\mathcal{W}}$ and the union of components from ${\mathcal{W}} ^*$.