On the generalized cocycle equation of Ebanks and Ng (Q1895188)

It is proved that the solution \(F_j : M^2 \to \Gamma\), \(j = 1, \dots, 6\), (where \(M\) is an abelian monoid and \(\Gamma\) is a uniquely 3-divisible abelian group) of a single functional equation considered by \textit{B. R. Ebanks} and \textit{C. T. Ng} [Aequationes Math. 46, No. 1-2, 76-90 (1993; Zbl 0801.39008)] may be decomposed into a sum of some functions \(G\) and \(H\) (mapping \(M^2\) into \(\Gamma)\) satisfying one of the following equations: \(\Delta_{xyz} G = \Delta_{yzx} G\) and \(\Delta_{xyz} H + \Delta_{yzx} H + \Delta_{zxy} H = 0\), where \(\Delta_{abc} K : = K(a + b,c) - K(a,c) - K(b,c)\).