Reversible Disjoint Unions of Well Orders and Their Inverses

Abstract

© 2019, Springer Nature B.V. A poset ℙ is called reversible iff every bijective homomorphism f: ℙ→ ℙ is an automorphism. Let W and W∗ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form ℙ=⋃i∈ILi, where Li, i∈ I, are pairwise disjoint linear orders from W∪ W∗. First, if Li∈ W, for all i ∈ I, and Li≅ αi= γi+ ni∈ Ord , where γi ∈Lim ∪{0} and ni ∈ ω, defining Iα := {i ∈ I : αi = α}, for α ∈Ord, and Jγ := {j ∈ I : γj = γ}, for γ ∈Lim ∪{0}, we prove that ⋃i∈ILi is a reversible poset iff 〈αi : i ∈ I〉 is a finite-to-one sequence, that is, |Iα| < ω, for all α ∈Ord, or there exists γ = max{γi : i ∈ I}, for α ≤ γ we have |Iα| < ω, and 〈ni : i ∈ Jγ ∖ Iγ〉 is a reversible sequence of natural numbers. The same holds when Li∈ 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 W and the union of components from W∗.