Yang–Baxter Solutions from Generalized Left Semi-Braces

Definitions and mathematical statement

Let $S$ be a generalized left semi-brace, with operations denoted by $+$ and $\circ$, and with the notation $a^{-}$ used in the supplied source. The record does not supply the defining axioms of a generalized left semi-brace or define this inverse notation; these must be fixed from the cited source before this formulation is fully self-contained.

Define the map $r\colon S\times S\to S\times S$ by \[ r(a,b)=\left(a\circ(a^{-}+b),(a^{-}+b)^{-}\circ b\right) \qquad(a,b\in S). \] Write $\operatorname{id}_S$ for the identity map on $S$, and define maps on $S^3$ by \[ r_{12}=r\times\operatorname{id}_S, \qquad r_{23}=\operatorname{id}_S\times r. \] The problem is to characterize precisely those generalized left semi-braces $S$ for which $r$ is a set-theoretic solution of the Yang--Baxter equation, expressed in braid form as \[ r_{12}\circ r_{23}\circ r_{12} = r_{23}\circ r_{12}\circ r_{23} \quad\text{as maps }S^3\longrightarrow S^3. \] Here composition between maps denotes ordinary function composition, whereas $\circ$ inside the formula for $r$ denotes the operation on $S$. The displayed map identity is required on every triple $(a,b,c)\in S^3$. The supplied record does not specify whether its term "set-theoretic solution" additionally requires bijectivity or nondegeneracy; any such requirement must be retained if imposed by the cited source. No finiteness hypothesis or additional restriction on $S$ is supplied.

Short English statement

Characterize the generalized left semi-braces for which the displayed map $r$ satisfies the Yang--Baxter equation. A fully self-contained statement requires confirmation of the semi-brace axioms, the meaning of $a^{-}$, and any additional requirements included in the source's definition of a solution.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30004313$, title: "Yang--Baxter Solutions from Generalized Left Semi-Braces"; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied report citation: Mini-Workshop: Algebraic Tools for Solving the Yang--Baxter Equation (2020), Oberwolfach Reports, pp. 3207--3242, DOI 10.4171/owr/2019/51. https://doi.org/10.4171/owr/2019/51. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Set-theoretic solutions to Yang--Baxter and generalized semi-braces. https://doi.org/10.1515/forum-2020-0082.
• Supplied literature reference: Semi-braces and the Yang--Baxter equation. https://www.sciencedirect.com/science/article/pii/S0021869317302284.
• Supplied literature reference: "Oberwolfach Reports, semi-brace problem." https://ems.press/content/serial-article-files/46831.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that necessary and sufficient criteria are known for narrower left semi-brace settings, sufficient constructions exist for generalized left semi-braces, and the requested full characterization is not known. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.