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

TL;DR

This paper explores conditions under which triples of subsets in groups satisfy the triple product property, impacting matrix multiplication embedding and providing bounds on subset sizes.

Contribution

It introduces elementary disjointness conditions for TPP triples, classifies disjointness types, and derives bounds on subset sizes within the TPP framework.

Findings

Nine disjointness case types identified

Four disjointness classes classified

Bounds on sum of subset sizes derived

Abstract

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.