ON THE COINCIDENCE OF THE FACTOR AND GONDRAN-MINOUX RANK FUNCTIONS OF MATRICES OVER A SEMIRING

摘要

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran-Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran-Minoux ranks are different.