  We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in
the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level
quantum system. General necessary and sufficient conditions for a mixed Weyl
quantum channel of an arbitrary dimension to be accessible by a semigroup are
established. The set ${\cal A}_N^Q$ is shown to be log--convex and star-shaped
with respect to the completely depolarizing channel. A decoherence supermap
acting in the space of Lindblad operators transforms them into the space of
Kolmogorov generators of classical semigroups. We show that for mixed Weyl
channels the hyper-decoherence commutes with the dynamics, so that decohering a
quantum accessible channel we obtain a bistochastic matrix form the set ${\cal
A}_N^C$ of classical maps accessible by a semigroup. Focusing on $3$-level
systems we investigate the geometry of the sets of quantum accessible maps, its
classical counterpart and the support of their spectra. We demonstrate that the
set ${\cal A}_3^Q$ is not included in the set ${\cal U}^Q_3$ of quantum
unistochastic channels, although an analogous relation holds for $N=2$. The set
of transition matrices obtained by hyper-decoherence of unistochastic channels
of order $N\ge 3$ is shown to be larger than the set of unistochastic matrices
of this order, and yields a motivation to introduce the larger sets of
$k$-unistochastic matrices.
