Some structure theorems on pseudo-symmetric sets
A pseudo-symmetric set is a binary system \((S,\mathbb{O})\) satisfying (1) \(a\mathbb{O}a=a\), (2) \((a\mathbb{O}b)\mathbb{O}c=(a\mathbb{O}c)\mathbb{O}(b\mathbb{O}c) \) and (3) the mapping \(\sigma_a: x\to x\mathbb{O}a\) is a permutation on \(S\). The group of automorphisms generated by all \(\sigma_a\) is denoted \(G(S)=G\). We set \(H(S)=<\sigma_a^{-1}\sigma_b \mid a,b\in S>\), the group of displacements of \(S\). There is a close connection between the structure of \(S\) and that of \(H(S)\). In this paper a structure theorem on solvable pseudo-symmetric sets is obtained: \(S\) is solvable (or nilpotent) iff \(H(S)\) is so (theorems 7 and 8). If \(L\) is a nilpotent Lie algebra of finite dimension and \(\sigma(a)=\exp(\operatorname{ad} a)\) for \(a\in L\), then \(L\) is considered as pseudo-symmetric set. Then \(L\) is nilpotent as pseudo-symmetric set.
scientific article; zbMATH DE number 3851176
scientific article