<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

Published Online: Feb 07, 2023

Page range: 49 - 66

Received: Oct 17, 2022

Accepted: Jan 10, 2023

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

Keywords
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

Published Online: Feb 07, 2023

Page range: 49 - 66

Received: Oct 17, 2022

Accepted: Jan 10, 2023

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

Keywordssine 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.

Keywords
sine subtraction law, semigroup, homomorphic involution, anti-homomorphic involution, topological semigroup