Sine subtraction laws on semigroups

Bruce Ebanks

Published: 2023-02-07

Vol. 37 No. 1 (2023)

Sine subtraction laws on semigroups

Bruce Ebanks

Abstract

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.

Keywords:

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

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Ebanks, B. (2023). Sine subtraction laws on semigroups. Annales Mathematicae Silesianae, 37(1), 49–66. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/15209

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References

J. Aczél and J. Dhombres, Functional Equations in Several Variables. With Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989.
Google Scholar

B. Ebanks, The cosine and sine addition and subtraction formulas on semigroups, Acta Math. Hungar. 165 (2021), no. 2, 337–354.
Google Scholar

B. Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups, Results Math. 77 (2022), no. 1, Paper No. 11, 14 pp.
Google Scholar

B. Ebanks and H. Stetkær, d’Alembert’s other functional equation on monoids with an involution, Aequationes Math. 89 (2015), no. 1, 187–206.
Google Scholar

H. Stetkær, Functional Equations on Groups, World Scientific, Singapore, 2013.
Google Scholar

H. Stetkær, Extensions of the sine addition law on groups, Aequationes Math. 93 (2019), no. 2, 467–484.
Google Scholar

H. Stetkær, Homomorphisms of product semigroups, Results Math. 77 (2022), no. 2, Paper No. 60, 21 pp.
Google Scholar

W.H. Wilson, On certain related functional equations, Bull. Amer. Math. Soc. 26 (1920), no. 7, 300–312.
Google Scholar

Vol. 37 No. 1 (2023)
Published: 2023-03-03

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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