Generalized Cayley's $\mathrm{\Omega }$-processes

In this paper we generalize some constructions and results due to Cayley and Hilbert. We define the concept of $\mathrm{\Omega }$–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 $\mathrm{\Omega }$-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.