  The Tsallis entropy given for a positive parameter $\alpha$ can be considered
as a modification of the classical Shannon entropy. For the latter,
corresponding to $\alpha=1$, there exist many axiomatic characterizations. One
of them based on the well-known Khinchin-Shannon axioms has been simplified
several times and adapted to Tsallis entropy, where the axiom of (generalized)
Shannon additivity is playing a central role. The main aim of this paper is to
discuss this axiom in the context of Tsallis entropy. We show that it is
sufficient for characterizing Tsallis entropy with the exceptions of cases
$\alpha=1,2$ discussed separately.
