From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures

The interplay between set-theoretic solutions of the Yang--Baxter equation Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body literature in recent years. In paper, Alan Koch generalised construction Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism group \cdot)$ whose image is abelian. this we characterise endomorphisms for which Koch's construction, variation on it, yield (bi-)skew braces. We show derived by carry over to our more general situation, discuss related structures.