Abstract—

In the paper, the elementary net closure problem is considered. An elementary net (net without a diagonal) σ = (σ_ij)_i ≠ j of additive subgroups σ_ij of field k is called “closed” if elementary net group E(σ) does not contain new elementary transvections. Elementary net σ = (σ_ij) is called “supplemented” if table (with a diagonal) σ = (σ_ij), 1 ≤ i, j ≤ n, is a (full) net for some additive subgroups σ_ii of field k. The supplemented elementary nets are closed. The necessary and sufficient condition for the supplementarity of elementary net σ = (σ_ij) is the implementation of inclusions σ_ijσ_jiσ_ij ⊆ σ_ij (for any i ≠ j). The question (Kourovka Notebook, Problem 19.63) is investigated of whether it true that, for closure of elementary net σ = (σ_ij) it suffices to implement inclusions \(\sigma _{{ij}}^{2}{{\sigma }_{{ji}}}\) ⊆ σ_ji for any i ≠ j (here, (\(\sigma _{{ij}}^{2}\) denotes the additive subgroup of field k generated by the squares from σ_ij). The elementary nets for which the latter inclusions are satisfied are called “weakly supplemented elementary nets.” The concepts of supplemented and weakly supplemented elementary nets coincide for fields of odd characteristic. Thus, the aforementioned question of the sufficiency of weak supplementarity for the closure of an elementary net is relevant for the fields of characteristics 0 and 2. In this paper, examples of weakly supplemented but not supplemented elementary nets are constructed for the fields of characteristics 0 and 2. An example of a closed elementary net that is not weakly supplemented is constructed.

中文翻译：

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关于封闭基本网的充分条件

摘要-

在本文中，考虑了基本的网络封闭问题。如果基本网组E（σ）不包含新的基本对流，则字段k的加性子组σij的基_本网（无对角线的网络）σ=（_σij）_{i  ≠  j}。初等净σ=（σ _IJ）被称为“补充”，如果表（对角线）σ=（σ _IJ），1≤我，Ĵ ≤ Ñ，是一种（全）净一些添加剂亚组σ_二场的ķ。补充的基本网是封闭的。对于基本净σ的替补=（σ的充分必要条件_IJ）是实施夹杂物σ _IJ σ_ジσ _IJ ⊆σ _IJ（对于任何我≠ Ĵ）。研究问题（Kourovka笔记本，问题19.63）是否成立，对于闭合基本网σ=（_σij）是否足以实现包含物\（\ sigma _ {{{ij}} ^ {2} {{\西格玛} _ {{ジ}}} \） ⊆σ_ジ任何我≠ Ĵ（这里，（\（\西格玛_ {{IJ}} ^ {2} \）表示字段的添加剂子组ķ由正方形从σ产生_IJ）。满足后一个包含的基本网称为“弱补充基本网”。补充和弱补充的基本网的概念在具有奇特特性的场中是一致的。因此，前面提到的弱补充足以满足基本网的封闭性的问题与特征0和2的领域有关。在本文中，构造了弱补充但未补充的基本网的示例，用于特征0的领域2.构造一个没有弱补充的封闭基本网的例子。