Summary: This paper is concerned to additive and multiplicative systems of homogeneous difference equations of non-negative degree. We apply a reduction in order for both additive and multiplicative systems. Then, we consider convergence and monotony of positive solutions. In fact, using convergence results on factor maps, we obtain convergence results on homogeneous systems. We will conclude that monotonic behaviour on the invariant ray (i.e. \(x=\bar r y\) for multiplicative systems and \(x=\bar s + y\) for additive systems) may or may not be the representative of other solutions. To illustrate our results, some examples are presented by multiplicative and additive homogeneous systems of rational equations.