The sine addition and subtraction formulas on semigroups
The functional equation \(f(xy) = f(x)g(y)+g(x)f(y),\,\, x,y \in S\) is called the sine addition formula on a semigroup \(S\). Several mathematicians have investigated its solutions on groups, regular semigroups, and semigroups which are generated by their squares. In the paper the author finds the general solution on all semigroups using a transfinite induction argument. In addition, he presents the general solution of the sine subtraction formula \(f(x\sigma(y))=f(x)g(y) - g(x)f(y)\) on monoids, where \(\sigma\) is an automorphic involution. He also gives the general continuous solutions on topological semigroups.
scientific article; zbMATH DE number 7377606