  We investigate the possibility of dividing quantum channels into
concatenations of other channels, thereby studying the semigroup structure of
the set of completely-positive trace-preserving maps. We show the existence of
'indivisible' channels which can not be written as non-trivial products of
other channels and study the set of 'infinitesimal divisible' channels which
are elements of continuous completely positive evolutions. For qubit channels
we obtain a complete characterization of the sets of indivisible and
infinitesimal divisible channels. Moreover, we identify those channels which
are solutions of time-dependent master equations for both positive and
completely positive evolutions. For arbitrary finite dimension we prove a
representation theorem for elements of continuous completely positive
evolutions based on new results on determinants of quantum channels and
Markovian approximations.
