Sine subtraction laws on semigroups - Ebanks, Bruce (Annales Mathematicae Silesianae, 2023)
We consider two variants of the sine subtraction law on a semigroup S. The main objective is to solve f(xy*) = f(x)g(y) - g(x)f(y) for unknown functions f,g: S→ℂ, where x↦x* is an anti-homomorphic involution. Until now this equation was not solved even when S is a non-Abelian group and x* = x-1. We find the solutions assuming that f is central. A secondary objective is to solve f(xσ(y)) = f(x)g(y) - g(x)f(y), where σ: S→S is a homomorphic involution. Until now this variant was solved assuming that S has an identity element. We also find the continuous solutions of these equations on topological semigroups.