  The classes of 1MP-inverses and MP1-inverses are recently introduced classes
of generalized inverses of complex matrix. Actually, they coincide with the
classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider
these inverses in the context of a ring with involution and prove that their
most important characterizations and properties remain true. We show that the
binary relations based on these inverses are in fact the well known left-star
and right-star partial orders. We extend these relations to the ring case,
connect them with the unified theory of partial order relations based on
generalized inverses and provide several properties. Finally, we indicate how
these results can be applied to bounded Hilbert space operators.
