序理论树中居间关系的公理化
Axiomatization of betweenness in order-theoretic trees
AI总结：
本文针对序理论树中的居间关系，提出了其一元二阶公理化，并证明拟树中的诱导居间关系可由一阶句子刻画，推广了先前连接树的结果。
摘要：
树的二元居间关系B(x,y,z)表示y位于x和z之间的唯一路径上。这一概念可以推广到序理论树，即满足任意节点的上方节点集合构成线性序的偏序集。在这种广义树中，两个节点之间的唯一“路径”可能包含无限多个节点。我们推广了先前文章中关于连接树居间关系的一些结果。连接树是任意两个节点都有最小上界的序理论树。其动机是为了方便地定义可数图的秩宽。我们将基于连接树居间关系的结构称为拟树。我们证明了拟树可由一阶句子公理化。本文中，我们获得了序理论树中居间关系的一元二阶公理化。我们还定义并比较了几种诱导居间关系，即不同种类的广义树中居间关系对节点集合的限制。我们证明了拟树中的诱导居间关系可由一阶句子刻画。证明使用了序理论树。
英文摘要：
The ternary betweenness relation of a tree, B(x,y,z) expresses that y is on the unique path between x and z. This notion can be extended to order-theoretic trees defined as partial orders such that the set of nodes larger than any node is linearly ordered. In such generalized trees, the unique "path" between two nodes can have infinitely many nodes. We generalize some results obtained in a previous article for the betweenness of join-trees. Join-trees are order-theoretic trees such that any two nodes have a least upper-bound. The motivation was to define conveniently the rank-width of a countable graph. We called quasi-tree the structure based on the betweenness relation of a join-tree. We proved that quasi-trees are axiomatized by a first-order sentence. Here, we obtain a monadic second-order axiomatization of betweenness in order-theoretic trees. We also define and compare several induced betweenness relations, i.e., restrictions to sets of nodes of the betweenness relations in generalized trees of different kinds. We prove that induced betweenness in quasi-trees is characterized by a first-order sentence. The proof uses order-theoretic trees.
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