Sober L-convex spaces and L-join-semilattices

With a complete residuated lattice L as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of L-convex spaces. We provide a specific construction for the sobrification of an L-convex space and demonstrate that the full subcategory of sober L-convex spaces is reflective in the category of L-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott L-convex structures on L-ordered sets. As an application of this type of sobriety, we obtain a characterization for the L-join-semilattice completion of an L-ordered set: an L-ordered set Q is an L-join-semilattice completion of an L-ordered set P if and only if the Scott L-convex space (Q, σ∗ (Q)) is a sobrification of the Scott L-convex space (P, σ∗ (P)).