Set-Theoretical Solutions of the \(n\)-Simplex Equation

The \(n\) -simplex equation was introduced by Zamolodchikovas a generalization of the Yang–Baxter equation which becomes the \(2\) -simplex equation in this terms. In the presentarticle, we suggest general approaches to construction of solutions of the \(n\) -simplex equation, describe certain types ofsolutions, and introduce an operation that allows us to construct, under certain conditions,a solution of the \((n+ m + k)\) -simplex equation from solutions of the \((n + k)\) -simplex equation and \((m + k)\) -simplex equation. We consider the tropicalizationof rational solutions and discuss its generalizations. We prove that a solution of the \(n\) -simplex equation on \(G\) can be constructed from solutions of this equationon \(H\) and \(K\) if \(G\) is an extension of a group \(H\) by a group \(K\) . We also find solutions of the parametricYang–Baxter equation on \(H\) with parameters in \(K\) . We introduce ternary algebras for studyingthe 3-simplex equation and present examples of such algebras that provide us with solutions ofthe 3-simplex equation. We find all elementary verbal solutions of the 3-simplex equation on a freegroup. \(||\)