  Recently, there has been a renewed interest in the quantification of
coherence or other coherence-like concepts within the framework of quantum
resource theory. However, rigorously defined or not, the notion of coherence or
decoherence has already been used by the community for decades since the advent
of quantum theory. Intuitively, the definitions of coherence and decoherence
should be the two sides of the same coin. Therefore, a natural question is
raised: how can the conventional decoherence processes, such as the von
Neumann-L\"{u}ders (projective) measurement postulation or partially dephasing
channels, fit into the bigger picture of the recently established theoretic
framework? Here we show that the state collapse rules of the von Neumann or
L\"{u}ders-type measurements, as special cases of genuinely incoherent
operations (GIO), are consistent with the resource theories of quantum
coherence. New hierarchical measures of coherence are proposed for the
L\"{u}ders-type measurement and their relationship with measurement-dependent
discord is addressed. Moreover, utilizing the fixed point theory for
$C^\ast$-algebra, we prove that GIO indeed represent a particular type of
partially dephasing (phase-damping) channels which have a matrix representation
based on the Schur product. By virtue of the Stinespring's dilation theorem,
the physical realizations of incoherent operations are investigated in detail
and we find that GIO in fact constitute the core of strictly incoherent
operations (SIO) and generally incoherent operations (IO) and the unspeakable
notion of coherence induced by GIO can be transferred to the theories of
speakable coherence by the corresponding permutation or relabeling operators.
