关于满足三元积性质的三组群子集的不交条件的简短说明
A Short Note on Disjointness Conditions for Triples of Group Subsets Satisfying the Triple Product Property
AI总结：
本文研究了满足三元积性质的三组群子集的不交条件，并推导了其大小之和的上下界。
摘要：
我们推导了一些关于满足所谓三元积性质（TPP）的三组群子集的简单成对不交和半不交条件。这种TPP性质允许群“实现”矩阵乘法，其维度大小等于子集的大小，子集作为输入矩阵的索引集，嵌入到群的正则代数中。我们为任意TPP三元组推导了九种不同的不交情况，并根据一个整数度量来分类这些情况，该度量衡量子集之间的成对不交程度。最后，我们推导了构成TPP三元组的子集大小之和的下界和上界，这是最初由Cohn和Umans为TPP三元组子集大小乘积推导的乘法界限的加法等价形式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
