Generalizations of Cayley's $\Omega$-process

In this paper we axiomatize some constructions and results due to Cayley and Hilbert. We define the concept of $\Omega$-process for an arbitrary affine algebraic monoid with zero and unit group $G$. In our situation 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 $S(V)^G$ that is large enough to guarantee its finite generation. Moreover, using complete reducibility, we give an explicit construction of all $\Omega$-processes for reductive monoids.