  This article analyzes the algebraic structure of the set of all quantum
channels and its subset consisting of quantum channels that have Holevo
representation. The regularity of these semigroups under composition of
mappings are analysed. It is also known that these sets are compact convex sets
and, therefore, rich in geometry. An attempt is made to identify generalized
invertible channels and also the idempotent channels. When channels are of the
Holevo type, these two problems are fully studied in this article. The
motivation behind this study is its applicability to the reversibility of
channel transformations and recent developments in resource-destroying
channels, which are idempotents. This is related to the coding-encoding problem
in quantum information theory. Several examples are provided, with the main
examples coming from pre-conditioner maps which assigns preconditioners to
matrices, in numerical linear algebra.Thus the known pre-conditioner maps are
viewd as a quantum-channel in finite dimentions.
