  This paper provides some new characterizations of the diamond partial order
for rectangular matrices by using properties of inner inverses, minus order,
and SVD decompositions. In addition, the recently introduced 1MP generalized
inverse and its dual are used to characterize the diamond partial order as a
${\cal G}$-based one. Finally, the one-sided star partial order is investigated
by using 1MP- and MP1-inverses.
