Reversible Disjoint Unions of Well Orders and Their Inverses

TL;DR

This paper characterizes when certain unions of well orders and their inverses are reversible posets, providing necessary and sufficient conditions based on the structure of their components.

Contribution

It offers a complete characterization of reversibility for unions of disjoint well orders and their inverses, extending previous understanding of reversible posets.

Findings

Reversibility depends on the finiteness of certain index sets.

Conditions involve the structure of ordinal sequences and natural numbers.

Reversibility of the entire union reduces to reversibility of its parts.

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 ${\mathcal{W}} ^*$.