# Orthogonal Countable Linear Orders

Orthogonal Countable Linear Orders Order https://doi.org/10.1007/s11083-018-9460-9 Christian Delhomme´ Received: 23 September 2015 / Accepted: 1 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Two linear orderings of a same set are perpendicular if every self-mapping of this set that preserves them both is constant or the identity. Two isomorphy types of linear orderings are orthogonal if there exist two perpendicular orderings of these types. Our main result is a characterisation of orthogonality to ω : a countably infinite type is orthogonal to ω if and only if it is scattered and does not admit any embedding into the chain of infinite classes of its Hausdorff congruence. Besides we prove that a countable type is orthogonal to ω + n (2 ≤ n<ω) if and only if it has infinitely many vertices that are isolated for the order topology. We also prove that atype τ is orthogonal to ω + 1 if and only if it has a decomposition of the form τ = τ +1+τ with τ or τ orthogonal to ω, or one of them finite 1 2 1 2 nonempty and the other one orthogonal to ω + 2. Since it was previously known that two http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Order Springer Journals

# Orthogonal Countable Linear Orders

Order, Volume OnlineFirst – May 29, 2018
39 pages

/lp/springer_journal/orthogonal-countable-linear-orders-ZilsaNT6Z6
Publisher
Springer Journals
Subject
Mathematics; Order, Lattices, Ordered Algebraic Structures; Discrete Mathematics; Algebra
ISSN
0167-8094
eISSN
1572-9273
D.O.I.
10.1007/s11083-018-9460-9
Publisher site
See Article on Publisher Site

### Abstract

Order https://doi.org/10.1007/s11083-018-9460-9 Christian Delhomme´ Received: 23 September 2015 / Accepted: 1 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Two linear orderings of a same set are perpendicular if every self-mapping of this set that preserves them both is constant or the identity. Two isomorphy types of linear orderings are orthogonal if there exist two perpendicular orderings of these types. Our main result is a characterisation of orthogonality to ω : a countably infinite type is orthogonal to ω if and only if it is scattered and does not admit any embedding into the chain of infinite classes of its Hausdorff congruence. Besides we prove that a countable type is orthogonal to ω + n (2 ≤ n<ω) if and only if it has infinitely many vertices that are isolated for the order topology. We also prove that atype τ is orthogonal to ω + 1 if and only if it has a decomposition of the form τ = τ +1+τ with τ or τ orthogonal to ω, or one of them finite 1 2 1 2 nonempty and the other one orthogonal to ω + 2. Since it was previously known that two

### Journal

OrderSpringer Journals

Published: May 29, 2018

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