Set-Theoretic Solutions of the Pentagon Equation

Abstract

A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map $s:S^2\rightarrow S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times {{{\,\mathrm{id}\,}}}$, $s_{23}={{{\,\mathrm{id}\,}}}\times s$ and $s_{13}=(\tau \times {{{\,\mathrm{id}\,}}})({{{\,\mathrm{id}\,}}}\times s)(\tau \times {{{\,\mathrm{id}\,}}})$ are mappings from $S^3$ to itself and $\tau :S^2\rightarrow S^2$ is the flip map, i.e., $\tau (x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2={{\,\mathrm{id}\,}}$. It is shown that such solutions are determined by a factorization of S as direct product $X\times A \times G$ and a map $\sigma :A\rightarrow {{\,\mathrm{Sym}\,}}(X)$, where X is a non-empty set and A, G are elementary abelian 2-groups. Isomorphic solutions are determined by the cardinalities of A, G and X, i.e., the map $\sigma$ is irrelevant. In particular, if S is finite of cardinality $2^n(2m+1)$ for some $n,m\geqslant 0$ then, on S, there are precisely $\left( {\begin{array}{c}n+2\\ 2\end{array}}\right)$ non-isomorphic solutions of the Pentagon Equation.