The paper presents descriptions of the solutions to the Hosszú functional equation \(f(x + y - xy) + f(xy) = f(x) + f(y)\) and its pexiderization \(\begin{aligned} f(x + y - xy) + g(xy) = h(x) + k(y), \end{aligned}\) in the class of maps from a field \(\mathbb {F}\) into an abelian cancellative semigroup. Next, as the main results, we show how to derive from them the form of the set-valued solutions of the equations and of the equation \(p_1f(q_1 x+q_2 y+q_3 xy)+ p_2f(q_4xy)= p_3f(q_5x)+ p_4f(q_6y)\) , with fixed \(q_1,\ldots ,q_6\in \mathbb {F}\setminus \{0\}\) . At the end of the paper some open problems are stated.
