Do all bracoids yield solutions to the Yang–Baxter equation?

Determine whether every bracoid (G, ·, N, *, odot) as defined by Martin-Lyons and Truman—namely, a quintuple with groups (G, ·) and (N, *), a transitive action odot of G on N, and satisfying the bracoid relation g odot (η * μ) = (g odot η) * (g odot e_N)^{-1} * (g odot μ) for all g in G and η, μ in N—necessarily gives rise to a set-theoretic solution of the Yang–Baxter equation.

Background

Braces are known to produce bijective, non-degenerate set-theoretic solutions to the Yang–Baxter equation, and the inverse solution is obtained by considering the opposite brace. Bracoids, introduced as a generalization of skew left braces, have been shown to yield solutions in certain instances.

Despite several constructions of bracoids (including those built from abelian maps and strong left ideals) and demonstrated right non-degenerate solutions under special conditions, it remains unresolved whether the bracoid framework always produces a solution to the Yang–Baxter equation. The paper highlights this gap while presenting special cases where solutions do arise.