  This paper studies several problems concerning channel inclusion, which is a
partial ordering between discrete memoryless channels (DMCs) proposed by
Shannon. Specifically, majorization-based conditions are derived for channel
inclusion between certain DMCs. Furthermore, under general conditions, channel
equivalence defined through Shannon ordering is shown to be the same as
permutation of input and output symbols. The determination of channel inclusion
is considered as a convex optimization problem, and the sparsity of the weights
related to the representation of the worse DMC in terms of the better one is
revealed when channel inclusion holds between two DMCs. For the exploitation of
this sparsity, an effective iterative algorithm is established based on
modifying the orthogonal matching pursuit algorithm.
