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≥3, ⌊d/2⌋, 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⌊d2/4⌋ 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,⋯,n, let pi be a prime number or 0, and let Vi be the 3-dimensional vector space over the prime field of characteristic pi. We prove that the direct product of the lattices Sub(V1),.., Sub(Vn) is 4-generated if and only if each of the numbers p1,.., pn occurs at most four times in the sequence p1,.., pn. Neither this direct product nor any of the subspace lattices Sub(V) above is 3-generated. © The Author(s), under exclusive licence to University of Szeged 2024.