Reversible Disjoint Unions of Well Orders and Their Inverses

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}$}$,forall$i\in I$,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 $\mathrm{\mid }{I}_{\alpha }\mathrm{\mid }<\omega$, and $\langle n_i : i\in J\gamma \setminus I\gamma \rangle $isareversiblesequenceofnaturalnumbers\mathrm{.}Thesameholdswhen${$\mathbb{L}$} _i \in {$\mathcal{W}$} ^*$,forall$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 ${\mathcal{W}}^{\ast }$.