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Motivated by reachability questions in coherently controlled open quantumsystems coupled to a thermal bath, as well as recent progress in the field ofthermo-/vector-majorization we generalize classical majorization from unitalquantum channels to channels with an arbitrary fixed point $D$ of full rank.Such channels preserve some Gibbs-state and thus play an important role in theresource theory of quantum thermodynamics, in particular inthermo-majorization.  Based on this we investigate $D$-majorization on matrices in terms of itstopological and order properties, such as existence of unique maximal andminimal elements, etc. Moreover we characterize $D$-majorization in the qubitcase via the trace norm and elaborate on why this is a challenging task whengoing beyond two dimensions.

Exploring the Limits of Open Quantum Dynamics II: Gibbs-Preserving Maps from the Perspective of Majorization