Orthogonal countable linear orders (Q2314417)

The introduction of the concept of perpendicular orders in the 1990-ties was motivated by the problem of description of the lattice of clones on a set. Two partial orders on a set are said to be perpendicular, if the only order-preserving self-maps preserving them both are the identity and constant maps. Two order types \(\alpha\) and \(\beta\) are said to be orthogonal if there is a pair of perpendicular orders of type \(\alpha\), resp., \(\beta\). Lastly, two orders are said to be orthogonal if so are their types. The authors have already shown in [Discrete Math. 341, No. 7, 1885--1899 (2018; Zbl 1435.06001)] that if two denumerable types of linear orders are both distinct from \(\omega + n\) and from the reversed type \(n + \omega\) for any \(n < \omega\), then they are orthogonal. In the paper under review, they characterise (separately) also those countable linear order types that are orthogonal to \(\omega\), to \(\omega + 1\), and to \(\omega + n\) for some (equivalently, for every) \(n \ge 2\). So, they have completed a characterisation of orthogonality of pairs of types of countable linear orderings. Given a countable chain \(\mathfrak C\), let \(\mathfrak C^\perp\) stand for the class of countable chains orthogonal to \(\mathfrak C\). Let \(\mathfrak C \sim \mathfrak D\) mean that every countable chain orthogonal to \(\mathfrak C\) iff it is orthogonal to \(\mathfrak D\). The above results infer that the equivalence relation \(\sim\) has 7 classes.