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Journal: Annales Mathematicae Silesianae

Sine subtraction laws on semigroups

Bruce Ebanks

Language: EN | Published: 07-02-2023 | Abstract | pp. 49-66

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.

2023-02-07



Journal: Annales Mathematicae Silesianae

Cosine and sine addition and subtraction law with an automorphism

Youssef Aserrar , Elhoucien Elqorachi

Language: EN | Published: 29-11-2023 | Abstract | pp. 155-176

Let S be a semigroup. Our main result is that we describe the complex-valued solutions of the following functional equations
g(xσ(y)) = g(x)g(y) + f(x)f(y), x, y ∈ S,
f(xσ(y)) = f(x)g(y) + f(y)g(x), x, y ∈ S,
and
f(xσ(y)) = f(x)g(y) - f(y)g(x), x, y ∈ S,
where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants.
We also give some applications.

2023-11-29