Yang–Baxter Solutions from Skew Lattices

Definitions and mathematical statement

A skew lattice is a set $S$ equipped with two associative, idempotent binary operations $\wedge$ and $\vee$ satisfying, for every $x,y\in S$, \[ x\wedge(x\vee y)=x=(y\vee x)\wedge x, \qquad x\vee(x\wedge y)=x=(y\wedge x)\vee x. \] Neither operation is required to be commutative.

For a set $X$, use the braid convention for a set-theoretic Yang--Baxter solution: a map $r\colon X\times X\to X\times X$ satisfies \[ (r\times\mathrm{id}_X)\circ(\mathrm{id}_X\times r)\circ(r\times\mathrm{id}_X) = (\mathrm{id}_X\times r)\circ(r\times\mathrm{id}_X)\circ(\mathrm{id}_X\times r) \] as maps $X^3\to X^3$. This definition alone imposes no bijectivity requirement; the supplied question does not specify whether bijectivity or other additional properties are intended.

Problem. Can skew lattices be used to describe further set-theoretic solutions of the Yang--Baxter equation, and what kinds of solutions arise from such descriptions?

The supplied formulation is an open-ended construction and characterization problem. It does not specify the previously known families relative to which solutions must be further, the permitted constructions from a skew lattice, whether $X$ must equal $S$ or may be an associated set, or additional hypotheses on the skew lattice or the solution. No finiteness, nonemptiness, invertibility, involutivity, or nondegeneracy condition is supplied. Consequently, a uniquely quantified classification or existence assertion cannot be extracted without further source clarification.

Short English statement

Investigate constructions of set-theoretic Yang--Baxter solutions from skew lattices and characterize the solutions they produce. The source leaves both the allowed constructions and the meaning of further solutions unspecified. Its open status is a supplied dated assessment requiring independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004314$, title: Yang--Baxter Solutions from Skew Lattices; exact source problem code: catalogue entry. The record belongs to Oberwolfach Reports Open Problems and supplies proposed year $2019$; no individual proposer is identified.
• Supplied source 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 supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Skew lattices and set-theoretic solutions of Yang--Baxter. https://doi.org/10.1016/j.jalgebra.2019.10.007.
• Supplied literature reference: Yang--Baxter semitrusses. https://arxiv.org/abs/2109.04978.
• Supplied literature reference: Generalized digroups and di-skew braces. https://link.springer.com/article/10.1007/s00233-025-10606-2.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that skew-lattice constructions characterize several families of set-theoretic Yang--Baxter solutions, while describing all further solutions obtainable from skew lattices remains unresolved. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.