Monotonically equivalent entropies and solution of additivity equation

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.