Axiomatizations Of Betweenness In Order-Theoretic Trees

The ternary betweenness relation of a tree, B (x , y, z), expresses that the node y is on the unique path between nodes 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 is linearly ordered and can be infinite.We generalize some results obtained in a previous article for the betweenness relation 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 (N, B) based on the betweenness relation B of a join-tree with vertex set N. We proved that quasi-trees are axiomatized by a first-order sentence.Here, we obtain a monadic second-order axiomatization of betweenness in ordertheoretic trees. We also define and compare several induced betweenness relations, i.e., restrictions to sets of nodes of the betweenness relations in countable 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.