Piecewise convex embeddability on linear orders

Given a nonempty set $L$ of linear orders, we say that the linear order L is $L$-convex embeddable into the linear order $L^{\prime }$ if it is possible to partition L into convex sets indexed by some element of $L$ 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 [13]), which are the special cases $L=\left\{1\right\}$ and $L=\mathrm{Fin}$. 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.