A partial ordering of discrete, memoryless channels

This paper is concerned with the structure of a partial ordering of discrete, memoryless communication channels. These are identified with equivalence classes of stochastic matrices into which the set of all stochastic matrices is partitioned by a relation of matrix inclusion. The relation carries over to channels and induces a partial ordering on them, having the property that if
K_{1}
and
K_{2}
are channels such that
K_{1}
includes
K_{2}
, then if a code exists for
K_{2}
, there exists a code for
K_{1}
, whose probability of error is never greater than that of the code for
K_{2}
. Results are derived which specify the equivalence classes of stochastic matrices corresponding to the binary and the symmetric channels and resolve the structure of the partial ordering between them.