Induced betweenness in order-theoretic trees

Betweenness is an abstract topological notion that has been studied for a long time in different structures. Informally, the ternary relation B(x, y, z) states that an element y is between x and z, in a sense that depends on the considered structure. In a partially ordered set (N, ≤), B(x, y, z):⇐⇒ x < y < z ∨ z < y < x. The corresponding betweenness structure is (N, B). The class of betweenness structures of linear orders is first-order definable; in other words, it is axiomatized by a first-order sentence. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order (N, ≤) such that, the set of elements larger that any element is linearly ordered, and any two elements have an upper-bound. A rooted tree T ordered by the ancestor relation is an order-theoretic tree. In an order-theoretic tree, we define B(x, y, z) to mean that x < y < z or z < y < x or x < y ≤ x⊔z or z < y ≤ x ⊔ z provided the least upper-bound x ⊔ z of x and z is defined when x and z are incomparable. In a previous article, we established that the corresponding class BO of betweenness structures is monadic second-order definable. We left as a conjecture that the class IBO of induced substructures of the structures in BO is monadic second-order definable. We prove this conjecture. Our proof uses partitioned probe cographs (related to cographs), and their six finite minimal excluded induced subgraphs called their bounds. This proof links two apparently unrelated topics: cographs and order-theoretic trees. However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not yet known. © 2022 by the author(s)