  We analyze the relation between CP-divisibility and the lack of information
backflow for an arbitrary -- not necessarily invertible -- dynamical map. It is
well known that CP-divisibility always implies lack of information backflow.
Moreover, these two notions are equivalent for invertible maps. In this letter
it is shown that for a map which is not invertible the lack of information
backflow always implies the existence of completely positive (CP) propagator
which, however, needs not be trace-preserving. Interestingly, for a {\em wide
class of image non-increasing dynamical maps} this propagator becomes
trace-preserving as well and hence the lack of information backflow implies
CP-divisibility. This result sheds new light into the structure of the
time-local generators giving rise to CP-divisible evolutions. We show that if
the map is not invertible then positivity of dissipation/decoherence rates is
no longer necessary for {CP-}divisibility.
