Reversible Disjoint Unions of Well Orders and Their Inverses

Abstract

A poset $\mathbb{P}$ is called reversible iff every bijective homomorphism $f:\mathbb{P}\to \mathbb{P}$ is an automorphism. Let $\mathcal{W}$ and ${\mathcal{W}}^{\ast }$ 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 $\mathcal{W}\cup {\mathcal{W}}^{\ast }$. 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 $\gamma \in Li{m}_0$, we prove that ${\bigcup }_{i\in I}{\mathbb{L}}_i$ is a reversible poset iff $⟨{\alpha }_i:i\in I⟩$ is a finite-to-one sequence, or there is $\gamma =\max \{{\gamma }_i:i\in I\}$, for $\alpha \le \gamma$ we have $|I_{\alpha }|<\omega$, and $⟨n_i:i\in J_{\gamma }\setminus I_{\gamma }⟩$ is a reversible sequence of natural numbers. The same holds when ${\mathbb{L}}_i\in {\mathcal{W}}^{\ast }$, 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 ${\mathcal{W}}^{\ast }$.

12 pages