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

TL;DR

This paper broadens the understanding of generating subspace lattices by bounding the minimal generating number $f^{ ext{mng}}(L)$ for $L= ext{Sub}(_F V)$ and its direct powers, relating these bounds to field generation $f^{ ext{mng}}(F)$ and the dimension $d$ via $M=ig floor d^2/4ig floor$. It extends Zádori’s 5-generation result to all 1- or 2-generated fields and generalizes Gelfand–Ponomarev’s findings, proving $f^{ ext{mng}}(L) eq 3$ and showing $f^{ ext{mng}}(L^k) o 5+m$ in many cases, with $m=ig ceil t/Mig ceil$ and $t=f^{ ext{mng}}(F)$. The work deploys coordinatization theory, projective-geometry representations, and coordinate rings to construct generating vectors, derives exact thresholds for direct products over prime fields, and provides constructive methods (including explicit matrices and lattice terms) for generating sets. Collectively, these results have implications for lattice generation and potential cryptographic considerations, and the authors supply Maple tools to reproduce key computations used in the proofs.

Abstract

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 $F$, $t$, $V$, $d\geq 3$, $[d/2]$, 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 $m[d^2/4]$ 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 $k$-th 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.

Figures (9)

• Figure 1: The 3-dimensional projective space
• Figure 2: Computing reciprocals
• Figure 3: For $\vec{g}=\vec{g}^{(i)}$, $\textup{typ}(\vec{g})$ cannot be $(2,2)$
• Figure 4: A quadruple of points not in general position
• Figure 5: Proving Fact \ref{['fact:tmgsfjtkgmnDhmkPsSr']}

Theorems & Definitions (26)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 3.5
• Example 3.6
• Remark 3.7
• Theorem 4.1: von Neumann bookvonneumann for $3\leq d\in{\mathbb N^+}$ and Day and Pickering daypick for $d=3$
• proof : Proof of Observation \ref{['obs:hbNvlnlgnD']}
• proof : Proof of Lemma \ref{['lemma:bgzsLm']}