Generating subspace lattices, their direct products, and their direct powers

In 2008, László Zádori proved that the lattice \(Sub (V) \) of all subspaces of a vector space V of finite dimension at least 3 over a finite field F has a 5-element generating set; in other words, \(Sub (V) \) is 5-generated. We prove that the same holds over every 1- or 2-generated field; in particular, over every field that is a finite degree extension of its prime field. Furthermore, let F, t, V, \(d\ge 3\), \(\lfloor d/2\rfloor \), and m denote an arbitrary field, the minimum cardinality of a generating set of F, a finite dimensional vector space over F, the dimension (assumed to be at least 3) of V, the integer part of d/2, and the least cardinal such that \(m\lfloor d^2/4\rfloor \) is at least t, respectively. We prove that \(Sub (V) \) is \((4+m)\)-generated but none of its generating sets is of size less than m. Moreover, the kth direct power of \(Sub (V) \) is \((5+m)\)-generated for many positive integers k; for all positive integers k if F is infinite. Finally, let n be a positive integer. For \(i=1,\dots , n\), let \(p_i\) be a prime number or 0, and let \(V_i\) be the 3-dimensional vector space over the prime field of characteristic \(p_i\). We prove that the direct product of the lattices \(Sub (V_1) \), ..., \(Sub (V_n) \) is 4-generated if and only if each of the numbers \(p_1\), ..., \(p_n\) occurs at most four times in the sequence \(p_1\), ..., \(p_n\). Neither this direct product nor any of the subspace lattices \(Sub (V) \) above is 3-generated.