  We investigate decompositions of Betti diagrams over a polynomial ring within
the framework of Boij--Soederberg theory. That is, given a Betti diagram, we
decompose it into pure diagrams. Relaxing the requirement that the degree
sequences in such pure diagrams be totally ordered, we are able to define a
multiplication law for Betti diagrams that respects the decomposition and
allows us to write a simple expression the decomposition of the Betti diagram
of any complete intersection in terms of the degrees of its minimal generators.
In the more traditional sense, the decomposition of complete intersections of
codimension at most 3 are also computed as given by the totally ordered
decomposition algorithm obtained from (Eisenbud-Schreyer, 2009). In higher
codimension, obstructions arise that inspire our work on an alternative
algorithm.
