# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Doyle, PH; Hocking, JG, Bijectively related spaces, I. Manifolds. Pac. J. Math., 111, 23-33 (1984) · Zbl 0554.57014  Dow, A.; Hernández-Gutiérrez, R., Reversible filters, Topology Appl., 225, 34-45 (2017) · Zbl 1368.54007  Kukieła, M., Reversible and bijectively related posets, Order, 26, 119-124 (2009) · Zbl 1178.06002  Kukieła, M., Characterization of hereditarily reversible posets, Math. Slovaca, 66,3, 539-544 (2016) · Zbl 1389.06002  Kurilić, MS, Retractions of reversible structures, J. Symb. Log., 82,4, 1422-1437 (2017) · Zbl 1423.03111  Kurilić, MS; Morača, N., Condensational equivalence, equimorphism, elementary equivalence and similar similarities, Ann. Pure Appl. Logic, 168,6, 1210-1223 (2017) · Zbl 1422.03062  Kurilić, M.S., Morača, N.: Reversibility of extreme relational structures, (submitted) arXiv:1803.09619  Kurilić, M.S., Morača, N.: Reversible sequences of cardinals, reversible equivalence relations, and similar structures, (submitted) arXiv:1709.09492 · Zbl 1422.03062  Kurilić, M.S., Morača, N.: Reversibility of disconnected structures, arXiv:1711.01426  Laflamme, C.; Pouzet, M.; Woodrow, R., Equimorphy: the case of chains, Arch. Math. Logic, 56, 7-8, 811-829 (2017) · Zbl 1417.06001  Laver, R., An order type decomposition theorem, Ann. Math., 98,1, 96-119 (1973) · Zbl 0264.04003  Rajagopalan, M.; Wilansky, A., Reversible topological spaces, J. Aust. Math. Soc., 61, 129-138 (1966) · Zbl 0151.29602  Rosenstein, JG, Linear Orderings Pure and Applied Mathematics, 98 (1982), New York: Academic Press, Inc., Harcourt Brace Jovanovich Publishers, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.