Let S be a semigroup, let M be a monoid with neutral element e, and let \(\mathbb {K}\) be an algebraically closed field of characteristic \(\ne 2\) with identity element 1. Inspired by Stetkær’s procedure [19] we describe, in terms of multiplicative functions and characters of 2-dimensional representations of S, the solutions \(g:S\rightarrow \mathbb {K}\) of the functional equation

where \(\psi :S\rightarrow S\) is an anti-endomorphism that need not be involutive and \(\mu :S\rightarrow \mathbb {K}\) is a multiplicative function such that \(\mu (x\psi (x))=1\) for all \(x\in S\). This enables us to find the solutions \(g:M\rightarrow \mathbb {K}\) of the new functional equation

where \(\sigma :M\rightarrow M\) is an involutive endomorphism.

设S是一个半群，设M是一个具有中性元素e的幺半群，并令\(\mathbb {K}\)是特征\(\ne 2\)的代数闭域，其单位元素为 1。受 Stetkær 程序的启发[19] 我们根据S的乘法函数和二维表示的特征描述了函数方程的解\(g:S\rightarrow \mathbb {K}\)

其中\(\psi :S\rightarrow S\)是一个不需要对合的反内同态，\(\mu :S\rightarrow \mathbb {K}\)是一个乘法函数，使得\(\mu (x \psi (x))=1\)对于所有\(x\in S\)。这使我们能够找到新函数方程的解\(g:M\rightarrow \mathbb {K}\)

其中\(\sigma :M\rightarrow M\)是对合内同态。