Betweenness in order-theoretic trees - Archive ouverte HAL Access content directly
Conference Papers Year : 2020

Betweenness in order-theoretic trees

Abstract

The ternary betweenness relation of a tree, B(x, y, z), indicates 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 greater 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 have called quasi-tree 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 between-ness 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.
Fichier principal
Vignette du fichier
CourcelleYuri2020.pdf (196.38 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-02497172 , version 1 (03-03-2020)

Identifiers

  • HAL Id : hal-02497172 , version 1

Cite

Bruno Courcelle. Betweenness in order-theoretic trees. Journées sur les Arithmétiques Faibles, May 2020, Fontainebleau, France. ⟨hal-02497172⟩

Collections

CNRS
28 View
123 Download

Share

Gmail Facebook X LinkedIn More