Quaternion-valued multiplicative functions on semigroups

Our aim is to solve a system of functional equations closely related to trigonometric functional equations. This allows us to express quaternion-valued multiplicative functions in terms of complex-valued multiplicative functions. As an application of our results, we give the continuous quaternion-valued solutions of a functional equation on \(({\mathbb {R}},\cdot )\) $(\mathbb{R},·)$ .