Reversible Disjoint Unions of Well Orders and Their Inverses

2017/11/19 by Kurilić, Miloš S., Morača, Nenad
#03C07 #03E10 #06A05 #06A06 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1711.07053

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 form ℙ =\bigcup _{i∈ I}\mathbbL _i, where \mathbbL _i, i∈ I, are pairwise disjoint linear orders from W ∪ W ^*. First, if \mathbbL _i ∈ W, for all i∈ I, and \mathbbL _i ≅ α_i =γ_i+n_i∈ 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∈ I} \mathbbL _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|

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