The $n$-simplex equation was introduced by Zamolodchikov as a generalization of the Yang-Baxter equation which becomes the 2-simplex equation in this terms. In the present article, we suggest general approaches to construction of solutions of the $n$-simplex equation, describe certain types of solutions, 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 tropicalization of 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 equation on $H$ and $K$ if $G$ is an extension of a group $H$ by a group $K$. We also found solutions of the parametric Yang-Baxter equation on $H$ with parameters in $K$. We introduce ternary algebras for studying the 3-simplex equation and present examples of such algebras that provide us with solutions of the 3-simplex equation. We found all elementary verbal solutions of the 3-simplex equation on a free group.

Representations of braid group $B_n$ on $n\ge2$ strands by automorphisms of a free group of rank $n$ go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group $VB_n$ on $n\ge2$ strands is an extension of the classical braid group $B_n$ by the symmetric group $S_n$. In this paper, we consider flat virtual braid groups $FVB_n$ on $n\ge2$ strands and construct a family of representations of $FVB_n$ by automorphisms of free groups of rank $2n$. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case $n=2$ and proved that the kernel contains a free group of rank two for $n\ge3$.