\(n\)-valued quandles and associated bialgebras

We study \(n\)-valued quandles and \(n\)-corack bialgebras. These structures are closely related to topological field theories in dimensions \(2\) and \(3\), to the set-theoretic Yang–Baxter equation, and to the \(n\)-valued groups, which have attracted considerable attention or researchers. We elaborate the basic methods of this theory, find an analogue of the so-called coset construction known in the theory of \(n\)-valued groups, and construct \(n\)-valued quandles using \(n\)-multiquandles. In contrast to the case of \(n\)-valued groups, this construction turns out to be quite rich in algebraic and topological applications. We study the properties of \(n\)-corack bialgebras, which play a role similar to that of bialgebras in group theory.