<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...

A Levi–Civita Equation on Monoids, Two Ways

Published Online: May 12, 2022

Page range: 151 - 166

Received: Dec 12, 2021

Accepted: Apr 23, 2022

DOI: https://doi.org/10.2478/amsil-2022-0009

Keywords
Levi–Civita equation, sine addition formula, cosine addition formula, semigroup, monoid, exponential function

© 2022 Bruce Ebanks, published by Sciendo

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

A Levi–Civita Equation on Monoids, Two Ways

Published Online: May 12, 2022

Page range: 151 - 166

Received: Dec 12, 2021

Accepted: Apr 23, 2022

DOI: https://doi.org/10.2478/amsil-2022-0009

KeywordsLevi–Civita equation, sine addition formula, cosine addition formula, semigroup, monoid, exponential function

© 2022 Bruce Ebanks, published by Sciendo

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

Keywords
Levi–Civita equation, sine addition formula, cosine addition formula, semigroup, monoid, exponential function