Our main objective is to solve the hexagonal functional equation \(\begin{aligned} F(x_1&\rho (y_1), x_2 \sigma (y_2)) - F(x_1 \rho (y_1), x_2) - F(x_1, x_2 \sigma (y_2))\\&= F(x_1 y_1, x_2 y_2) - F(x_1 y_1, x_2) - F(x_1, x_2 y_2) \end{aligned}\) for an unknown \(F: S_1\times S_2 \rightarrow H\) , where \(S_1,S_2\) are monoids with respective involutions \(\rho \) and \(\sigma \) , and H is an Abelian group. Our method uses results about the solutions \(f_1,f_2,f_3:S \rightarrow H\) of \(\begin{aligned} f_1(xy)+ f_2(x\sigma (y)) = f_3(x), \quad x,y \in S, \end{aligned}\) which is a Pexiderized version of the extended Jensen equation on semigroups. We are motivated by some earlier results about similar functional equations on groups. In those results the equation is considered for \(S_1 = S_2 = G\) , a group, and with \(H = {\mathbb {C}}\) . Chung et al. (J. Korean Math. Soc. 38, 37-47 (2001)) solved the main equation under the assumptions that G is a 2-divisible group and \(\rho (x) = \sigma (x) = x^{-1}\) for all \(x \in G\) . Later Hunt et al. (Aequat. Math. 90, 87-96 (2016)) studied the special case \(\rho = \sigma \) of our equation on a general group, and Lin et al. (Aequat. Math. 97, 639-648 (2023)) studied our equation on a general group. The authors of the latter two articles claim that they solved the respective equations without assuming that the group S is 2-divisible, but some solutions are missing. We correct the latter two results and extend all three results to the case that S is a monoid.
