Decomposition of elementary transvection in elementary net group

The paper deals with the study of elementary nets (carpets) σ=(σij) and elementary net groups E(σ). Namely, decomposition of an elementary transvection in elementary net group E(σ) is given. The colections of subsets (ideals, additive subgroups and etc.) σ={σij:1≤i,j≤n} of an associative ring with the conditions σirσrj⊆σij, 1≤i,r,j≤n, arose in a different situations. Such collections are called carpets or nets and a rings, while the associated groups are called carpet (net, congruence, etc.) subgroups. An elementary net (a net without diagonal) σ is closed (admissible) if the subgroup E(σ) does not contain new elementary transvections. The study was motivated by the question of V. M. Levchuk (The Kourovka notebook, question 15.46) whether or not a necessary and sufficient condition for the admissibility (closure) of the elementary net σ is the admissibility (closure) of all pairs (σij,σji). In other words, the inclusion of an elementary transvection tij(α) in the elementary group E(σ) is equivalent to the inclusion of tij(α) in the subgroup ⟨tij(σij),tji(σji)⟩ (for any i≠j)...