A variety $X$ over a field $K$ is of Hilbert type if $X\left(K\right)$ is not thin. We prove that if $f:X\to S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^{-1}\left(s\right)$, $s\in S\left(K\right)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

We investigate the lattice $𝐋\left(𝐕\right)$ of subspaces of an $m$-dimensional vector space $𝐕$ over a finite field $\mathrm{GF}\left(q\right)$ with a prime power $q=p^n$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋\left(𝐕\right)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋\left(𝐕\right)$ is orthomodular. For...