Reversible Disjoint Unions of Well Orders and Their Inverses

Abstract

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 formwhere 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 i∈I L i is a reversible poset iff α i : i ∈ I is a finite-to-one sequence, or there is 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 L i ∈ 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 * . 2010 MSC: 06A06, 06A05, 03E10, 03C07.

Nonreversible trees having a removable edge

“…Then there is the sequence z n : n ∈ ω of different elements from Root X \{ f n (x * ) : n ∈ ω} such that z 0 = z * and f −1 [{z k }] = {z k+1 } for all k ∈ ω. Since |ρ \ [ρ]| = 1, by (8) and (9) we conclude that z k+1 ∈ Iso z k+1 ↑ , z k ↑ , for all k ∈ ω, which means that…”

“…By (8) and (11) (8) and (9) we conclude that x n ∈ Iso x n ↑ , x n+1 ↑ , for all n ∈ Z \ {0}, which means that…”

“…Similarly, f n (x * ) ∈ Iso( f n (x * ) ↑ , f n+1 (x * ) ↑ ) for n < m, and, in particular, x * ↑ w * ↑ . By (8) and (11), we have…”

“…1 (8), (9), and since x ↑ is connected that |ρ \ [ρ]| > 1. We conclude that in this case, by removing the edge u, v from X, the connectivity component x * ↑ was split into two components.…”

“…• f −1 [{x * }] ∩ { f n (x * ) : n ∈ ω} = ∅. Then there are sequences x n : n ∈ ω , y n : n ∈ ω , and z n : n ∈ ω of different elements from Root X , such that x 0 = y 0 = z 0 = x * , and such that (8) and (9) we conclude that x k ∈ Iso(x k ↑ , x k+1 ↑ ), for all k ∈ ω, and that y k+1 ∈ Iso(y k+1 ↑ , y k ↑ ), z k+1 ∈ Iso(z k+1 ↑ , z k ↑ ), for all k ∈ N, i.e.…”