Finite-valuation approximable structures: a solution to the Jung--Tix problem of probabilistic powerdomains

2026-08-03

AI Summary

This work resolves the long-standing Jung–Tix problem in domain theory: whether there exists a subcategory of continuous domains that is cartesian closed and closed under (sub)probabilistic powerdomains. To this end, we introduce the category of finitely valuation-approximable domains (FVA), constructed via FS-approximations of the identity on finite posets. By leveraging coreflective embeddings and a finite separation saturation theorem, we prove that FVA is closed under function spaces, finite products, Scott-continuous retracts, and (sub)probabilistic powerdomains. This constitutes the first demonstration of a subcategory of continuous domains simultaneously enjoying cartesian closure and closure under probabilistic powerdomains, thereby affirmatively settling the generalized Jung–Tix problem and confirming the well-behaved nature of the valuation monad within this setting.

Abstract

We introduce the category \(\FVA\) of finite-valuation approximable domains, a full subcategory of continuous domains contained in the category of pointed countably based FS-domains. We prove that \(\FVA\) is Cartesian closed and closed under both the subprobability and probability valuation powerdomains. Hence the valuation monads \(\Vsub\) and \(\Vone\) restrict to \(\FVA\), yielding a positive answer to the generalized form of Jung--Tix problem, one of the longest-standing open problem in domain theory since 1990s. The proof is divided into two steps. First, for every finite poset \(P\), we construct an increasing FS approximate identity on \(\Vsub(P)\), and thereby show that \(\Vsub(P)\) is a countably based FS-domain. Second, we call a domain finite-valuation approximable when its identity is the pointwise supremum of an increasing sequence of maps factoring through spaces \(\Vsub(P_n)\), where each \(P_n\) is finite. A finite-separation saturation theorem and a unified kernel-lifting theorem then show that \(\FVA\) is closed under Scott-continuous retracts, finite products, function spaces, \(\Vsub\), and \(\Vone\).

Problem

• probabilistic powerdomains
• Jung--Tix problem
• continuous domains
• valuation monads
• domain theory

Innovation

• finite-valuation approximable domains
• probabilistic powerdomains
• Jung–Tix problem
• Cartesian closed category
• valuation monads