Some Levi-Civita functional equations on semigroups (Q2152600)

Suppose that \(S\) is a semigroup. The functional equation \[ f(xy) =\sum_{i=1}^n g_i(x)h_i(y),\quad \forall \, x, y \in S, \] for \(n\in \mathbb{N}\) and unknown functions \(f, g_1, \dots , g_n, h_1,\dots, h_n : S\to \mathbb{C}\) is called the Levi-Civita equation. Recently, the author proved a general result giving explicit solution formulas for the Levi-Civita equation in the case \(n = 2\). In this paper, he gives explicit solution formulas in the case \(n =3\) on topological commutative monoids. Note that the complications caused by prime ideals increase significantly with respect to the case \(n = 2\). The paper also contains applications to specific functional equations and examples on a couple of commutative monoids with very different prime ideal structures.