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We deduce some elementary pairwise disjointness and semi-disjointnessconditions on triples of subsets in arbitrary groups satisfying the so-calledtriple product property (TPP) as originally defined by H. Cohn and C. Umans in2003. This property TPP for a triple of group subsets, called a TPP triple,allows the group to "realize" matrix multiplication of dimensions the sizes ofthe subsets, with the subsets acting as indexing sets for input matrices whichare embedded into the regular algebra of the group. We derive nine differentdisjointness casetypes for an arbitrary TPP triple, and classify these intofour different disjointness classes based on an integer measure of the degreeof pairwise disjointness among the subsets. Finally, we derive lower and upperbounds for the sum of sizes of the subsets forming a TPP triple, which is theadditive equivalent of the multiplicative bounds originally derived by Cohn andUmans for the product of sizes of subsets forming a TPP triple.

A Short Note on Disjointness Conditions for Triples of Group Subsets Satisfying the Triple Product Property