Example 2

\* Let A be a ring with an identity element, and for each $\alpha \in I$ let $T_\alpha$ be an Artinian left A-module. Let $E_\alpha$ be a homogeneous space for $T_\alpha$ on which $T_\alpha$ operates faithfully (so that $E_\alpha$ is an affine space attached to $T_\alpha$). For $\beta \geqslant \alpha$, suppose that $f_{\alpha\beta} : E_\beta \to E_\alpha$ is an affine mapping. Take $\mathfrak{S}_\alpha$ to be the set consisting of the empty set and the affine linear varieties in $E_\alpha$. Then condition (i) is trivially satisfied, and (ii) follows from the fact that $T_\alpha$ is Artinian; for this implies that there exists a minimal element in the set of finite intersections of sets $M \in \mathfrak{F}$, and this minimal element must be equal to $\bigcap_{M \in \mathfrak{F}} M$. Finally, since $f_{\alpha\beta}$ is affine, conditions (iii) and (iv) are trivially satisfied. \*