发表机构
School of Mathematics, Sichuan University（四川大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在$[0,1]$-富足范畴框架下研究模糊数空间上模糊序的完备性，刻画了三种连续三角模下的$S$-收敛性，并给出Cauchy完备与Smyth完备的充要条件。
AI 中文摘要
本文在$[0,1]$-富足范畴框架下研究了模糊数空间$\mathbb{E}^1$上模糊序的完备性性质。刻画了在三种基本连续三角模下的$S$-收敛性，并建立了诱导开球拓扑之间的包含关系。证明了$(\nmathbb{E}^1,P)$是Cauchy完备的当且仅当三角模不同构于Łukasiewicz三角模，而$(\nmathbb{E}^1,P)$对每个连续三角模都不是Smyth完备的。因此，对称化$(\nmathbb{E}^1,S)$是Smyth完备的当且仅当底层连续三角模不同构于Łukasiewicz三角模。
英文摘要
This paper investigates the completeness properties of the fuzzy order on the fuzzy number space $\mathbb{E}^1$ within the framework of $[0,1]$-enriched categories. \(S\)-convergence under the three fundamental continuous t-norms is characterized, and inclusion relations among the induced open ball topologies are established. It is shown that $(\mathbb{E}^1,P)$ is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas $(\mathbb{E}^1,P)$ fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization $(\mathbb{E}^1,S)$ is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.
