Document Type : Research Paper
Authors
Nanjing University of Information Science and Technology
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 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. {\color{blue} 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 $(P, \sigma^{\ast}(P))$.
Keywords
Main Subjects
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