Set theoretical solutions of equations of n -- simplexes

The n-simplex equation (n-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang-Baxter equation, which is, in these terms, a 2-simplex equation. In this article we propose some general approaches to constructing solutions to equations of $n$-simplices, describe some types of solutions, and introduce an operation that, under certain conditions, allows us to construct a solution (n + m + k)-SE from solutions ( n + k)-SE and (m + k)-SE. We consider tropicalization of rational decisions and discuss ways to generalize it. We prove that if G is an extension of H by K, then we can find a solution of n-SE on G from the solutions of this equation on H and K. Also, we find solutions to the parametric Yang-Baxter equation on H with parameters from K. To study the 3-simplex equation, we introduced algebraic systems with ternary operations and gave examples of these systems that give 3-SE solutions. We find all elementary solutions of 3-SE on a free group.

УДК 512.56

Keywords: Yang-Baxter equation, tetrahedral equation, $n$-simplex equation, set-theoretic solution, groupoid, 2-groupoid, ternary, ternoid, group extension, virtual braid group