$n$-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.