有限赋值可逼近结构：概率幂域的Jung–Tix问题的一个解决方案
Finite-valuation approximable structures: a solution to the Jung--Tix problem of probabilistic powerdomains
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- Sichuan University（四川大学）
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中文总结 AI 辅助
本文引入有限赋值可逼近域范畴FVA，证明其笛卡尔闭且在两类赋值幂域下封闭，解决了域论中自90年代起的Jung–Tix问题的广义形式。
中文摘要 AI 辅助
我们引入有限赋值可逼近域的范畴\boldsymbol{\text{FVA}}，它是连续域的满子范畴，包含在带点的可数基FS-域的范畴中。我们证明\boldsymbol{\text{FVA}}是笛卡尔闭的，且在子概率赋值幂域和概率赋值幂域下封闭。因此，赋值单子\boldsymbol{\text{Vsub}}和\boldsymbol{\text{Vone}}限制到\boldsymbol{\text{FVA}}，为Jung–Tix问题的广义形式给出肯定答案，该问题是自20世纪90年代以来域论中最悠久的开放问题之一。证明分为两步：第一步，对每个有限偏序集\boldsymbol{P}，我们在\boldsymbol{\text{Vsub}(P)}上构造一个递增的FS逼近恒等，从而证明\boldsymbol{\text{Vsub}(P)}是可数基FS-域；第二步，我们定义一个域为有限赋值可逼近，当且仅当其恒等是通过空间\boldsymbol{\text{Vsub}(P_n)}分解的映射的递增序列的逐点上确界，其中每个\boldsymbol{P_n}是有限的。随后，有限分离饱和定理和统一核提升定理表明，\boldsymbol{\text{FVA}}在Scott-连续收缩、有限积、函数空间、\boldsymbol{\text{Vsub}}和\boldsymbol{\text{Vone}}下封闭。
英文摘要
We introduce the category $ω{\bf FVA}$ of finite-valuation approximable domains, a full subcategory of continuous domains contained in the category of pointed countably based FS-domains. We prove that $ω{\bf FVA}$ is Cartesian closed and closed under both the subprobabilistic and probabilistic valuation powerdomains. Hence the valuation monads $\mathcal{V}_{\leq 1}$ and $\mathcal{V}_1$ restrict to $ω{\bf FVA}$, yielding a positive answer to the category-existence form of the Jung--Tix problem, a long-standing open problem in domain theory since the 1990s. In particular, we develop a new factorization-approximation framework for constructing objects from a given class of known objects or structures. Applying this method to the class of subprobabilistic powerdomains over finite posets, we construct the class $ω{\bf FVA}$ and show that it is closed under Scott-continuous retracts, finite products, function spaces, $\mathcal{V}_{\leq 1}$ and $\mathcal{V}_1$ monads.
