Given a semigroup S generated by its squares and equipped with an involutive automorphism \(\sigma \) and a multiplicative function \(\mu :S\rightarrow \mathbb {C}\) such that \(\mu (x\sigma (x))=1\) for all \(x\in S\), we determine the complex-valued solutions of the following functional equation

$$\begin{aligned} f(xy)-\mu (y)f(\sigma (y)x)=g(x)h(y),\quad x,y\in S. \end{aligned}$$

中文翻译：

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关于半群的d'Alembert泛函方程的推广

给定一个由其正方形生成的半群S并配备对合的自同构\（\ sigma \）和一个乘法函数\ {\ mu：S \ rightarrow \ mathbb {C} \）使得\（\ mu（x \ sigma（ x））= 1 \）对于所有\（x \ in S \），我们确定以下函数方程的复值解

$$ \ begin {aligned} f（xy）-\ mu（y）f（\ sigma（y）x）= g（x）h（y），\ quad x，y \ in S. \ end {aligned} $$