  In this article we explore several aspects concerning to the Moore-Penrose
inverse of a bounded linear operator. On the one hand, we study monotonicity
properties of the Moore-Penrose inverse with respect to the L\"owner, star,
minus, sharp and diamond orders. On the other hand, we analyze the validity of
the reverse order law, $B^\dagger A^\dagger=(AB)^\dagger$, under hypothesis of
operator ranges and also under hypothesis of order operators. Finally, we study
the operator $B^\dagger A^\dagger$ as different weighted inverses of $AB$.
