Piecewise convex embeddability on linear orders

Abstract

Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L^{\prime }$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$ which are isomorphic to convex subsets of $L^{\prime }$ ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in arXiv:2309.09910), which are the special cases $\mathcal{L}=\{1\}$ and $\mathcal{L}=\mathsf{Fin}$. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.

34 pages