Quaternion-valued multiplicative functions on semigroups (Q6657289)

Let \(S\) be a semigroup and let \(\mathbb{H}\) denote the skew field of the quaternions. The main aim of this paper is to determine quaternion-valued central multiplicative functions in terms of complex-valued multiplicative functions. To do this, at first they solve the system of functional equations\N\[\Na(xy)= a(x)a(y)-b(x)\overline{b(y)}, \qquad\Nb(xy)= a(x)b(y)+b(x)\overline{a(y)},\N\]\N\(x, y\in S\), where \(a, b \colon S \to\mathbb{C}\) denote the unknown functions and \(a\) is assumed to be central.