1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
access type Open Access

A Levi–Civita Equation on Monoids, Two Ways

Published Online: 12 May 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 12 Dec 2021
Accepted: 23 Apr 2022
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
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.

Keywords

MSC 2010

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