On sufficient conditions for the closure of an elementary net

Abstract

An elementary net (net without a diagonal) σ = (σij )i =j of additive subgroups σij of the field k is called closed if the elementary net group E(σ) does not contain new elementary transvections. Elementary net σ = (σij) is called supplemented if for some additive subgroups σii of the field k the table (with diagonal) σ = (σij ), 1 ≤ i, j ≤ n, is a (full) net. Supplemented elementary nets are closed. A necessary and sufficient condition for the supplementarity of the elementary net σ = (σij) is the implementation of inclusions σijσjiσij ⊆ σij (for any i = j). In this regard the following question is of interest (Kourovka notebook, question 19.63): ): is it true that for closedness of an elementary net σ = (σij) it suffices to execute the inclusions σ2 ijσji ⊆ σij for any i = j (here, by σ2 ij we denote the additive subgroup of the field k generated by all squares from σij)? Elementary nets for which the last inclusions are satisfied we call 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 closedness of an elementary net is relevant for fields of characteristic 0 and 2. In this article we construct examples of weakly supplemented but not supplemented elementary nets. An example of a closed elementary net is constructed, which is not weakly supplemented.