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Reversible disjoint unions of well orders and their inverses. (English) Zbl 07204025
Summary: 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 \(i \in I\), and \(\mathbb{L}_i \cong \alpha_i =\gamma_i+n_i\in \text{Ord} \), where \(\gamma_i \in \text{Lim} \cup \{0\}\) and \(n_i \in \omega \), defining \(I_\alpha := \{i \in I : \alpha_i = \alpha\}\) for \(\alpha \in \text{Ord}\), and \(J_\gamma := \{j \in I : \gamma_j = \gamma\}\), for \(\gamma \in \text{Lim} \cup\{0\}\), we prove that \(\bigcup_{i\in I} \mathbb{L}_i\) is a reversible poset iff \(\langle \alpha_i : i \in I \rangle\) is a finite-to-one sequence, that is, \(|I_\alpha| < \omega \), for all \(\alpha \in \text{Ord}\), or there exists \(\gamma = \max\{ \gamma_i :i \in I\}\), for \(\alpha \leq \gamma\) we have \(|I_\alpha| < \omega \), and \(\langle n_i : i \in J_\gamma \setminus I_\gamma \rangle\) is a reversible sequence of natural numbers. The same holds when \(\mathbb{L}_i \in \mathcal{W}^* \), for all \(i \in I\). 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 \(\mathcal{W}^* \).

06-XX Order, lattices, ordered algebraic structures
Full Text: DOI
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