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 ∈ I, and $\mathbb {L}_{i} \cong \alpha _{i} =\gamma _{i}+n_{i}\in \text {Ord}$, where γ_i ∈Lim ∪{0} and n_i ∈ ω, defining I_α := {i ∈ I : α_i = α}, for α ∈Ord, and J_γ := {j ∈ I : γ_j = γ}, for γ ∈Lim ∪{0}, we prove that $\bigcup _{i\in I} \mathbb {L}_{i}$ is a reversible poset iff 〈α_i : i ∈ I〉 is a finite-to-one sequence, that is, |I_α| < ω, for all α ∈Ord, or there exists γ = max{γ_i : i ∈ I}, for α ≤ γ we have |I_α| < ω, and 〈n_i : i ∈ J_γ ∖ I_γ〉 is a reversible sequence of natural numbers. The same holds when $\mathbb {L}_{i} \in \mathcal {W}^{*}$, for all i ∈ 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 $\mathcal {W}^{*}$.