Abstract
This paper concerns two notions of rank of matrices over semirings: semiring rank and column rank. These two rank functions are the same over fields and Euclidean rings, but differ for matrices over many combinatorially interesting semirings including the nonnegative integer matrices, the fuzzy matrices, and the Binary Boolean matrices. We investigate the largest value of r for which the column rank and semiring rank of all m× n matrices over a given semiring are both r. This value is determined for the semirings mentioned above as well as many others.
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