Sober $L$-凸空间与$L$-并半格
Sober $L$-convex spaces and $L$-join-semilattices
AI总结：
本研究将经典凸空间的sober性推广至$L$-凸空间，构造其sober化并证明范畴反射性，引入Scott $L$-凸结构，借此刻画$L$-序集的$L$-并半格完备化。
摘要：
以完备剩余格$L$为真值表，我们将经典凸空间的sober性定义推广到$L$-凸空间框架中。我们给出了$L$-凸空间sober化的具体构造，证明了sober $L$-凸空间的全子范畴在以保凸映射为态射的$L$-凸空间范畴中是反射的。此外，我们在$L$-序集上引入了Scott $L$-凸结构的概念。作为这类sober性的一个应用，我们得到了$L$-序集的$L$-并半格完备化的一个刻画：$L$-序集$Q$是$L$-序集$P$的$L$-并半格完备化当且仅当Scott $L$-凸空间$(Q, σ^{\u2217}(Q))$是Scott $L$-凸空间$(P, σ^{\u2217}(P))$的sober化。
英文摘要：
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 $(Q, σ^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, σ^{\ast}(P))$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
