The semigroup of prevarieties of linear group representations

Pairs $(G,\Gamma)$ are considered, where $G$ is a vector space over an arbitrary fixed field and $\Gamma$ is a group for which there is defined a representation with respect to $G$. A class of such pairs is called a prevariety if it is saturated and closed under the operations of forming subpairs and Cartesian products. A prevariety is called bounded if the variety generated by it differs from the class of all pairs. A prevariety is called small if it is generated by a single pair. The following theorems are proved.
Theorem 1. Over any field the semigroup of bounded prevarieties of pairs is free.
Theorem 2. The small prevarieties of pairs are indecomposable and generate a free semigroup of prevarieties.
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