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 \(L'\) if it is possible to partition \(L\) into convex sets, indexed by some element of \(L\), which are isomorphic to convex subsets of \(L'\) ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability, which arise from the special cases \(\mathcal{L}=\{\mathbf{1}\}\) and \(\mathcal{L}=\mathsf{Fin}\). In this talk we focus on the behaviour of these relations on the set of countable linear orders, first characterising when they are transitive, and hence a quasi-order. We then look at some combinatorial properties and complexity (with respect to Borel reducibility) of these quasi-orders. Finally, we analyse their extension to uncountable linear orders.

The presented results stem from joint work with Alberto Marcone, Luca Motto Ros, and Vadim Weinstein.