  We adopt the perspective of similarity equivalence, in gate set tomography
called the gauge, to analyze various properties of quantum operations belonging
to a semigroup, $\Phi= e^{{\cal L}t}$,and therefore given through the Lindblad
operator. We first observe that the non unital part of the channel decouples
from the time evolution. Focusing on unital operations we restrict our
attention to the single-qubit case, showing that the semigroup embedded inside
the tetrahedron of Pauli channels is bounded by the surface composed of product
probability vectors and includes the identity map together with the maximally
depolarizing channel. Consequently, every member of the Pauli semigroup is
unitarily equivalent to a unistochastic map, describing a coupling with
one-qubit environment initially in the maximally mixed state, determined by a
unitary matrix of order four.
