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We present connections between left non-degenerate solutions of set-theoreticYang-Baxter equation and left shelves using certain maps called Drinfel'dhomomorphisms. We further generalise the notion of affine quandle, by usingheap endomorphisms and metahomomorphisms, and identify the Yang-Baxter algebrafor solutions of the braid equation associated to a given quandle. We introducethe notion of the pre-Lie skew brace and identify certain affine quandles thatgive rise to pre-Lie skew braces. Generalisations of the braiding of a group,associated to set-theoretic solutions of the braid equation is also presented.These generalized structures encode part of the underlying Hopf algebra.Indeed, we also introduce the quasi-triangular Hopf algebras and the universalR-matrices for quandle algebras and for set-theoretic Yang-Baxter algebras. Infact, we obtain the universal R-matrix for the set-theoretic Yang-Baxteralgebras after identifying the associated admissible Drinfel'd twist. Genericset-theoretic solutions coming from heap endomorphisms are also identified.

Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices