Sufficient conditions for memory kernel master equation

We derive sufficient conditions for the memory kernel which guarantee legitimate (completely positive and trace-preserving) dynamical map. It turns out that these conditions provide a natural parameterizations of the dynamical map being a generalization of Markovian semigroup. It is shown that this class of maps cover almost all known examples -- from Markovian semigroup, semi-Markov evolution up to collision models and their generalization.