On the t-Term Rank of a Matrix

TL;DR

This paper generalizes the concept of term rank in (0,1)-matrices to t-term rank, providing formulas for maximum ranks and revealing the existence of matrices that realize all intermediate maximum ranks.

Contribution

It extends basic results of term rank to t-term rank and proves the existence of matrices achieving all intermediate maximum ranks within certain classes.

Findings

Derived a formula for maximum t-term rank in matrices with fixed row and column sums.

Proved the existence of matrices that realize all maximum ranks from 1 to t.

Generalized classical results from term rank to t-term rank.

Abstract

For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.