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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 $$\mathcal{W}^*$$.

##### MSC:
 06-XX Order, lattices, ordered algebraic structures
##### Keywords:
partial order; well order; disconnected structure; reversibility
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##### References:
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