Sober L-convex spaces and L-join-semilattices

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Wu G. j.; Yao W.

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Journal Paper

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Journal: Iranian Journal of Fuzzy Systems Year:2024 | Volume:21 | Issue:4 Page(s): 163-177

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Information Journal Paper

Title

Sober L-convex spaces and L-join-semilattices

Author(s)

Wu G. j. | Yao W. | Issue Writer Certificate

Keywords

$L$-convex spaces

sobriety

Scott $L$-convex structure

$L$-ordered set

$L$-join-semilattice

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 $(P, \sigma^{\ast}(P))$.

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