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

We consider the Levi–Civita equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0009_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>g</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>h</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>g</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>h</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math>
<tex-math>f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right)</tex-math>
</alternatives>
</disp-formula>
for unknown functions <italic>f, g</italic>1, <italic>g</italic>2, <italic>h</italic>1, <italic>h</italic>2 : <italic>S →</italic> ℂ, where <italic>S</italic> is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that <italic>f</italic> is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function <italic>f</italic> is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.
</abstract>

We consider the Levi–Civita equation



f(xy)=g1(x)h1(y)+g2(x)h2(y)
f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right)


for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

A Levi–Civita Equation on Monoids, Two Ways

Annales Mathematicae Silesianae

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

{"article-title":"A Levi\u2013Civita Equation on Monoids, Two Ways"}

We consider the Levi–Civita equation



f(xy)=g1(x)h1(y)+g2(x)h2(y)
f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x...

单体上的Levi–Civita方程，双向

在线发布:2022年5月12日

页面范围： 151 - 166

收到:2021年12月12日

认可的:2022年4月23日

内政部： https://doi.org/10.2478/amsil-2022-0009（网址：https://doi.org/10.2478/amsil-2022-0009）

关键词
Levi–Civita方程,正弦加法公式,余弦加法公式,半群,幺半群,指数函数

©2022 Bruce Ebanks，Sciendo出版

本作品根据知识共享署名4.0国际许可证授权。