Factor and term ranks for matrix union over semirings

We consider the arithmetic properties of the factor-rank and term-rank functions for matrices over semirings. In particular, we investigate the sets of matrices that satisfy the extremes of inequalities for these rank functions of matrix union. The classification of the linear transformations that keep these sets invariant is obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 175–197, 2003.