  We deduce some elementary pairwise disjointness and semi-disjointness
conditions on triples of subsets in arbitrary groups satisfying the so-called
triple product property (TPP) as originally defined by H. Cohn and C. Umans in
2003. This property TPP for a triple of group subsets, called a TPP triple,
allows the group to "realize" matrix multiplication of dimensions the sizes of
the subsets, with the subsets acting as indexing sets for input matrices which
are embedded into the regular algebra of the group. We derive nine different
disjointness casetypes for an arbitrary TPP triple, and classify these into
four different disjointness classes based on an integer measure of the degree
of pairwise disjointness among the subsets. Finally, we derive lower and upper
bounds for the sum of sizes of the subsets forming a TPP triple, which is the
additive equivalent of the multiplicative bounds originally derived by Cohn and
Umans for the product of sizes of subsets forming a TPP triple.
