Pairs of orthogonal countable ordinals

Abstract

Abstract. We characterize pairs of orthogonal countable ordinals. Two ordinals α and β are orthogonal if there are two linear orders A and B on the same set V with order types α and β respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if α and β are two countable ordinals, with α ≤ β, then α and β are orthogonal if and only if either ω + 1 ≤ α or α = ω and β < ωβ.

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

“…The results of this paper have been presented in part at the workshop on Homogeneous Structures, Banff, Nov. [8][9][10][11][12][13]2015 and to the seminar on Discrete Mathematics, Hamburg, Feb. 19. 2016.…”

Graphs containing finite induced paths of unbounded length

“…The proofs were based on uniformly recurrent sequences. We use similar ideas in [15] to prove that there are 2 ℵ 0 linear orders L of order type ω on N such that L is orthogonal to the natural order on N (see Theorem 2 and Corollary 2 of [15])…”

On Scattered Convex Geometries

“…Several papers related to bichains have appeared during the last few years. Some are about infinite bichains and are mostly concerned by their endomorphisms ( [42], [29], [19] This defines an order on the class S of all permutations. Classes C of permutations such that σ ∈ C whenever σ ≤ π for some π ∈ C are called hereditary.…”

“…Several papers related to bichains have appeared during the last few years. Some are about infinite bichains and are mostly concerned by their endomorphisms ( [42], [29], [19]). Many are about finite bichains and originate in the study of classes of permutations.…”