  Nonextensive entropies are divided into three classes, each of which is
characterized by Shannon additivity and pseudoadditivity. One of the three
classes has properties of both additivities. The remaining classes have only
one property of the two additivities, respectively. An example of nonextensive
entropy is shown concretely for each class. In particular, one class is found
to consist of only the Tsallis entropy. More precisely, the Tsallis entropy is
proved to be uniquely determined by only these two additivities. The present
classification using these two distinct additivities reveals unique
characteristics of the Tsallis entropy.
