Supernilpotence need not imply nilpotence (Q2314990)

The concept of higher commutator \([\alpha_0,\dots,\alpha_{ n-1}]\) is introduced for congruences \(\alpha_0,\dots,\alpha_ {n-1}\) on an algebra \(A\). Furthermore, \([\alpha]_0=\alpha=(\alpha]_0\) and \((\alpha]_{n+1}=[\alpha,(\alpha]_n]\) are defined. An algebra \(A\) is called \(n\)-\textit{step nilpotent} if \((\alpha]_n = 0\) for each \(\alpha\) in \(\mathrm{Con}(A)\). \(A\) is \(n\)-\textit{step supernilpotent} if \([\alpha,\dots,\alpha]=0\) (\(n-1\) times). Many important structural properties are shown to be associated with supernilpotence and the exact relationship between nilpotence and supernilpotence is investigated. The authors constract an example which is not solvable and hence not nilpotent but which is supernilpotent.