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 verbal solutions of 3-SE on a free group.