  In this paper we generalize some constructions and results due to Cayley and
Hilbert. We define the concept of $\Omega$--process for an arbitrary algebraic
monoid with zero and unit group $G$. Then we show how to produce from the
process and for a linear rational representation of $G$, a number of elements
of the ring of $G$-invariants, that is large enough as to guarantee its finite
generation. Moreover, we give an explicit construction of all
$\Omega$-processes for general reductive monoids and, in the case of the monoid
of all the $n^2$ matrices, compare our construction with Cayley's definition.
