We extend the classical notions of translativity and homogeneity of means to -homogeneity, that is, invariance with respect to an operation We find the shape of for the arithmetic weighted mean and then the general form of for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by -homogeneity with respect to a single operation , one needs to take two of such operations in order to characterize a mean.