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- W3017907887 abstract "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." @default.
- W3017907887 created "2020-05-01" @default.
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- W3017907887 date "2021-02-01" @default.
- W3017907887 modified "2023-10-14" @default.
- W3017907887 title "Axiomatization of betweenness in order-theoretic trees" @default.
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- W3017907887 doi "https://doi.org/10.23638/lmcs-17(1:11)2021" @default.
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