Sine subtraction laws on semigroups

Bruce Ebanks


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 xx* 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: SS 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.


sine subtraction law; semigroup; homomorphic involution; antihomomorphic involution; topological semigroup

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Published : 2023-02-07

EbanksB. (2023). Sine subtraction laws on semigroups. Annales Mathematicae Silesianae, 37(1), 49-66. Retrieved from

Bruce Ebanks
Department of Mathematics and Statistics, Mississippi State University  United States

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