Sober $L$-convex spaces and $L$-join-semilattices

TL;DR

This paper extends the concept of sobriety to $L$-convex spaces using a complete residuated lattice $L$, introduces Scott $L$-convex structures, and characterizes $L$-join-semilattice completions via sobrification.

Contribution

It develops a framework for sobriety in $L$-convex spaces, constructs sobrifications, and characterizes $L$-join-semilattice completions through Scott $L$-convex structures.

Findings

Sobrification of $L$-convex spaces is constructed explicitly.

Full subcategory of sober $L$-convex spaces is reflective.

Characterization of $L$-join-semilattice completions via sobrification.

Abstract

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, demonstrating 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 $(Q, \sigma^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, \sigma^{\ast}(P))$.