The Kraft sum as a monotone function on the refinement-ordered set of uniquely decipherable codes

The set of all uniquely decipherable (UD) codes is partially ordered by refinement, meaning that all strings in the cruder code can be represented as concatenations of strings taken from the finer code. The Kraft sum is a monotone (increasing) function on this poset. In the refinement order, chains of UD codes having the same Kraft sum are necessarily of the simple sequence type. A characterization of UD codes in terms of equalities involving Kraft sums is also given.