Hasse-Graph Descriptions of Valuation Semigroups

Definitions and mathematical statement

The supplied record refers to a weighted Hasse graph, whose directed edges are cover relations $p\gtrdot q$, with edge weight denoted by $b=b(p,q)$. It also refers to objects $\widehat Y_p$, rational function fields $K(\widehat Y_p)$, functions $f_p$ and $f_{pq}$, and a valuation semigroup $\Gamma_C$. The underlying partially ordered set, the geometric objects, the base field, the permitted weights, the datum $C$, the valuation defining $\Gamma_C$, and the proposed graph-based description of that semigroup are not defined in the supplied record.

Conjecture as recorded, with unresolved notation. The original statement asks whether the description of $\Gamma_C$ using only weighted-Hasse-graph data is valid whenever \[ \text{for every cover relation }p\gtrdot q\text{ of weight }b, \quad \exists\,\eta\in K(\widehat Y_p) \quad\text{such that}\quad \eta^{b}=f_{pq} \quad\text{in }K(\widehat Y_p). \] The existential quantifier is separate for each edge; the supplied wording imposes no additional compatibility condition between these choices.

The supplied clean statement instead writes the edge condition as \[ \text{for every cover relation }p\gtrdot q\text{ of weight }b, \quad \exists\,\eta\in K(\widehat Y_p) \quad\text{such that}\quad \eta^{b}=\frac{f_p}{f_q} \quad\text{in }K(\widehat Y_p). \] The record does not define $f_{pq}$ or identify it with $f_p/f_q$. Nor does it specify how this quotient is interpreted in $K(\widehat Y_p)$, including the necessary nonvanishing of its denominator there. Consequently, these two displayed hypotheses cannot be asserted to be equivalent on the supplied evidence.

The intended conclusion is the validity of the source's weighted-Hasse-graph description of $\Gamma_C$; that description is not supplied as a formula or construction. Thus the record supports this conditional conjecture schema, but does not supply enough definitions or a sufficiently explicit conclusion to state a self-contained semigroup equality. These unresolved qualifications require mathematical review.

Short English statement

For each weighted cover relation, the conjecture assumes that a specified rational function has a root of order equal to the edge weight, and asks whether this ensures the proposed graph description of $\Gamma_C$. The record omits that description and its geometric setup, and uses two right-hand sides without establishing their equality. Its open status and verification labels are supplied curation, not independently verified conclusions.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record catalog; supplied permanent ID $30004270$, exact source problem code catalogue entry, title Hasse-Graph Descriptions of Valuation Semigroups. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied source citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/46, pp. 2869--2909. https://doi.org/10.4171/owr/2019/46. The record assigns proposed year 2019; the relationship between that year and the citation's 2020 date is not explained.
• Supplied literature source: "OWR 2019 46," https://ems.press/content/serial-article-files/46825.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, says that the semigroup conjecture is explicitly stated in the report, and reports finding no later proof or counterexample resolving the stated general condition. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports a check against the official Oberwolfach report and claims preservation of its hypotheses, notation, and target. This is a supplied verification claim; no source visit or independent verification is asserted here.