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 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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Notes
- 1.
Betweenness in partial orders, work in preparation.
- 2.
Defined for a relational structure: two elements are adjacent if they belong to some tuple of some relation.
- 3.
An ordered tree is a rooted tree such that the set of sons of any node is linearly ordered. This notion is extended in [4] to join-trees. Ordered join-trees should not be confused with order-theoretic trees, that we call O-trees for simplicity.
- 4.
Formal definition in [5].
- 5.
This definition can be used in partial orders. The corresponding notion of betweenness is axiomatized in [7]. We will not use it for defining betweenness in order-theoretic trees, although these trees are defined as partial orders.
- 6.
The three cases of A(x, y, z) are exclusive by A2 and A3.
- 7.
By reference to Language Theory where words, terms and trees are transformed by transductions. There are strong links between language theoretical and logically defined transductions, see [6].
References
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Acknowledgement
I thank the referee for comments helping me to clarify many points.
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Courcelle, B. (2020). Betweenness in Order-Theoretic Trees. In: Blass, A., Cégielski, P., Dershowitz, N., Droste, M., Finkbeiner, B. (eds) Fields of Logic and Computation III. Lecture Notes in Computer Science(), vol 12180. Springer, Cham. https://doi.org/10.1007/978-3-030-48006-6_6
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DOI: https://doi.org/10.1007/978-3-030-48006-6_6
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