A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ 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 $\tau\colon S^2\to S^2$ 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 $\sigma\colon A\to\mathrm{Sym}(X)$, 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 $n,m\geq 0$ then, on $S$, there are precisely $\binom{n+2}{2}$ non-isomorphic solutions of the Pentagon Equation.

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五边形方程的集合论解

五边形方程在非空集 $S$ 上的集合论解是 $s\colon S^2\to S^2$ 的映射，使得 $s_{23}s_{13}s_{12}= s_{12}s_{23}$，其中 $s_{12}=s\times\mathrm{id}$、$s_{23}=\mathrm{id}\times s$ 和 $s_{13}=( \tau\times\mathrm{id})(\mathrm{id}\times s)(\tau\times\mathrm{id})$ 是从 $S^3$ 到自身和 $\tau\colon S^ 的映射2\to S^2$是翻转图，即$\tau (x,y) =(y,x)$。我们给出所有对合解的描述，即$s^2=\mathrm{id}$。结果表明，这样的解是由 $S$ 作为直接积 $X\times A \times G$ 和映射 $\sigma\colon A\to\mathrm{Sym}(X)$ 的因式分解确定的，其中 $ X$ 是一个非空集，$A,G$ 是基本的阿贝尔$2$-组。同构解由$A$、$G$ 和$X$ 的基数决定，即映射$\sigma$ 无关紧要。特别是，