It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω n of alln x n doubly stochastic matrices. The regular matrices in Ω n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω n . The paper is concluded with extensions of some of these results to the convex setS n of alln x n nonnegative matrices having row and column sums at most one.