<abstract xmlns="http://www.w3.org/1999/xhtml">

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

We consider two variants of the sine subtraction law on a semi-group 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.

Sine Subtraction Laws on Semigroups

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Sine Subtraction Laws on Semigroups"}

We consider two variants of the sine subtraction law on a semi-group S. The main objective is to solve f(xy∗ ) = f(x)g(y) − g(x)f(y) for unknown functions f, g...

Sine Subtraction Laws on Semigroups

Publié en ligne: 07 févr. 2023

Pages: 49 - 66

Reçu: 17 oct. 2022

Accepté: 10 janv. 2023

DOI: https://doi.org/10.2478/amsil-2023-0002

Mots clés
sine subtraction law, semigroup, homomorphic involution, anti-homomorphic involution, topological semigroup

© 2023 Bruce Ebanks, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Sine Subtraction Laws on Semigroups

Publié en ligne: 07 févr. 2023

Pages: 49 - 66

Reçu: 17 oct. 2022

Accepté: 10 janv. 2023

DOI: https://doi.org/10.2478/amsil-2023-0002

Mots cléssine subtraction law, semigroup, homomorphic involution, anti-homomorphic involution, topological semigroup

© 2023 Bruce Ebanks, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
sine subtraction law, semigroup, homomorphic involution, anti-homomorphic involution, topological semigroup