On the Kapranov ranks of tropical matrices (Q417453)

The author investigates the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field he shows that the rank-product inequality holds for the Kapranov rank, and he proves that the Kapranov rank respects Green's preorders on the semigroup of tropical \(n\)-by-\(n\) matrices. The rank-product inequality is shown to fail for the Kapranov rank over any finite ground field. He provides an example of a 7-by-7 0-1-matrix whose Kapranov rank is independent of the ground field, equals 6, and exceeds tropical rank.