Do all skew bracoids yield Yang–Baxter solutions

Determine whether every skew bracoid (G, ·, N, ★, ⊙), as defined by Martin–Lyons and Truman, gives rise to a set-theoretic solution to the Yang–Baxter equation without imposing additional constraints on the bracoid structure.

Background

Skew bracoids are a generalization of skew left braces introduced to expand the algebraic framework connecting group-theoretic structures with applications such as Hopf–Galois theory and the Yang–Baxter equation. It is well-known that every brace canonically yields a bijective, non-degenerate solution to the Yang–Baxter equation, but the situation for bracoids is more subtle.

This paper constructs families of bracoids from abelian maps and exhibits special cases that produce right non-degenerate solutions to the Yang–Baxter equation. Despite these advances, the authors explicitly note that it remains unknown whether every bracoid yields any solution to the Yang–Baxter equation, highlighting a foundational open problem in the theory.