On extension of the solutions of the Popoviciu type equations on groups (Q519933)

Let \((G,+)\) and \((H,+)\) be commutative groups and \(S\) be a subsemigroup of \(G\). The author shows that, under some reasonable restrictions on \(G\), \(H\), and \(S\), every solution \(f\colon S\to H\) of the Popoviciu type functional equation \[ M\Bigl(\frac{x+y+z}{m}\Bigr)+f(x)+f(y)+f(z)= N\left(f\Bigl(\frac{x+y}{n}\Bigr)+f\Bigl(\frac{y+z}{n}\Bigr)+f\Bigl(\frac{z+x}{n}\Bigr)\right) \] can be uniquely extended to the solution of the equation on \(G\). Besides some auxiliary results, the main tool of the proof is the Kuratowski-Zorn lemma.