  In earlier work, my colleagues and I developed a formalism for using
information theory to understand scales of organization and structure in
multi-component systems. One prominent theme of that work was that the
structure of a system cannot always be decomposed into pairwise relationships.
In this brief communication, I refine that formalism to address recent examples
which bring out that theme in a novel and subtle way. After summarizing key
points of earlier papers, I introduce the crucial new concept of an ancilla
component, and I apply this refinement of our formalism to illustrative
examples. The goals of this brief communication are, first, to show how a
simple scheme for constructing ancillae can be useful in bringing out
subtleties of structure, and second, to compare this scheme with another recent
proposal in the same genre.
