  Generalized entropies are studied as Lyapunov functions for the Master
equation (Markov chains). Three basic properties of these Lyapunov functions
are taken into consideration: universality (independence of the kinetic
coefficients), trace-form (the form of sum over the states), and additivity
(for composition of independent subsystems). All the entropies, which have all
three properties simultaneously and are defined for positive probabilities, are
found. They form a one-parametric family.
  We consider also pairs of entropies $S_{1}$, $S_{2}$, which are connected by
the monotonous transformation $S_{2}=F(S_{1})$ (equivalent entropies). All
classes of pairs of universal equivalent entropies, one of which has a
trace-form, and another is additive (these entropies can be different one from
another), were found. These classes consist of two one-parametric families: the
family of entropies, which are equivalent to the additive trace-form entropies,
and the family of Renyi-Tsallis entropies.
