Ranks of 0-1 arrays of size 2 x 2 x 2 and 2 x 2 x 2 x 2

TL;DR

This paper uses computer algebra to determine the ranks and canonical forms of 2x2x2 and 2x2x2x2 arrays with entries in {0,1} across different algebraic structures, providing detailed classification results.

Contribution

It provides the first comprehensive rank classification and canonical forms for these small arrays over multiple algebraic settings.

Findings

Ranks of arrays in different algebraic contexts are explicitly determined.

Canonical forms of arrays under group actions are identified.

Results enhance understanding of tensor ranks in small multidimensional arrays.

Abstract

We use computer algebra to determine the ranks of arrays of size 2 x 2 x 2 and 2 x 2 x 2 x 2 with entries in the set {0, 1} regarded as a field with two elements, as a Boolean algebra, and as non-negative integers. In the field case we also determine the canonical forms of the arrays with respect to the action of the direct product of the general linear groups.