The Yang–Baxter and pentagon equations are two well-known equations of Mathematical Physic. If S is a set, a map $$s:S\times S\rightarrow S\times S$$ s : S × S → S × S is said to be a set-theoretical solution of the quantum Yang–Baxter equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, \end{aligned}$$ s 23 s 13 s 12 = s 12 s 13 s 23 , where $$s_{12}=s\times {{\,\mathrm{id}\,}}_S$$ s 12 = s × id S , $$s_{23}={{\,\mathrm{id}\,}}_S\times s$$ s 23 = id S × s , and $$s_{13}=({{\,\mathrm{id}\,}}_S\times \tau )\,s_{12}\,({{\,\mathrm{id}\,}}_S\times \tau )$$ s 13 = ( id S × τ ) s 12 ( id S × τ ) and $$\tau $$ τ is the flip map, i.e., the map on $$S\times S$$ S × S given by $$\tau (x,y)=(y,x)$$ τ ( x , y ) = ( y , x ) . Instead, s is called a set-theoretical solution of the pentagon equation if $$\begin{aligned} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}. \end{aligned}$$ s 23 s 13 s 12 = s 12 s 23 . The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang–Baxter equation. Specifically, we present a new construction of solutions of the Yang–Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.

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半群上杨-巴克斯特方程和五边形方程的集合论解

Yang-Baxter 方程和五边形方程是数学物理的两个著名方程。如果 S 是一个集合，则映射 $$s:S\times S\rightarrow S\times S$$ s : S × S → S × S 被称为量子杨-巴克斯特方程的集合论解，如果$$\begin{aligned} s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, \end{aligned}$$ s 23 s 13 s 12 = s 12 s 13 s 23 ，其中 $$s_{12}=s\times {{\,\mathrm{id}\,}}_S$$ s 12 = s × id S , $$s_ {23}={{\,\mathrm{id}\,}}_S\times s$$ s 23 = id S × s ，并且 $$s_{13}=({{\,\mathrm{id}\ ,}}_S\times \tau )\,s_{12}\,({{\,\mathrm{id}\,}}_S\times \tau )$$ s 13 = ( id S × τ ) s 12 ( id S × τ ) 和 $$\tau $$ τ 是翻转映射，即 $$S\times S$$ S × S 上的映射由 $$\tau (x,y)=(y, x)$$ τ ( x , y ) = ( y , x ) 。相反，如果 $$\begin{aligned} s_{23}\, s 被称为五边形方程的集合论解，s_{13}\, s_{12}=s_{12}\, s_{23}。\end{对齐}$$ s 23 s 13 s 12 = s 12 s 23 。这项工作的主要目的是展示五边形方程的解如何成为获得 Yang-Baxter 方程新解的有用工具。具体来说，我们提出了一种新构造的 Yang-Baxter 方程的解，涉及五边形方程的两个特定解。为此，我们提供了一种在两个半群的匹配乘积上获得五边形方程解的方法，该半群是一个包含经典 Zappa 乘积的半群。我们提出了一种新构造的 Yang-Baxter 方程的解，涉及五边形方程的两个特定解。为此，我们提供了一种在两个半群的匹配乘积上获得五边形方程解的方法，该半群是一个包含经典 Zappa 乘积的半群。我们提出了一种新构造的 Yang-Baxter 方程的解，涉及五边形方程的两个特定解。为此，我们提供了一种在两个半群的匹配乘积上获得五边形方程解的方法，该半群是一个包含经典 Zappa 乘积的半群。