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

期刊论文
发表日期：

2024年6月26日

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Gábor Czédli

摘要

In 2008, László Zádori proved that the lattice $Sub (V) $ $S u b (V)$ 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) $ $S u b (V)$ 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, $d\ge 3$ $d \geq 3$ , $\lfloor d/2\rfloor $ $⌊ d / 2 ⌋$ , 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\lfloor d^2/4\rfloor $ $m ⌊ d^{2} / 4 ⌋$ is at least t, respectively. We prove that $Sub (V) $ $S u b (V)$ is $(4+m)$ $(4 + m)$ -generated but none of its generating sets is of size less than m. Moreover, the kth direct power of $Sub (V) $ $S u b (V)$ is $(5+m)$ $(5 + m)$ -generated for many positive integers k; for all positive integers k if F is infinite. Finally, let n be a positive integer. For $i=1,\dots , n$ $i = 1, \dots, n$ , let $p_i$ $p_{i}$ be a prime number or 0, and let $V_i$ $V_{i}$ be the 3-dimensional vector space over the prime field of characteristic $p_i$ $p_{i}$ . We prove that the direct product of the lattices $Sub (V_1) $ $S u b (V_{1})$ , ..., $Sub (V_n) $ $S u b (V_{n})$ is 4-generated if and only if each of the numbers $p_1$ $p_{1}$ , ..., $p_n$ $p_{n}$ occurs at most four times in the sequence $p_1$ $p_{1}$ , ..., $p_n$ $p_{n}$ . Neither this direct product nor any of the subspace lattices $Sub (V) $ $S u b (V)$ above is 3-generated.