Supernilpotent groups and \(3\)-supernilpotent loops (Q6601865)

There are two fundamental approaches how to generalize nilpotency of groups for other classes of algebras: central nilpotency and supernilpotency. In some varieties (like groups or rings) these notions coincide, however, for Mal'tsev varieties, we can only say that supernilpotency is stronger than central nilpotency.\N\NLoops are a Mal'tsev variety that seems to be close to groups, however there is a \(6\)-element example that is centrally nilpotent and not supernilpotent. Moreover, for a finite loop, computing its level of central nilpotency is algorithmically easy but computing the level of supernilpotency is extremely difficult so far.\N\NDeciding whether a loop is \(1\)-supernilpotent or \(2\)-supernilpotent is obvious, hence the first difficult step comes with \(3\)-supernilpotency. In this paper the authors develop an approach using commutators and associators to construct an equational basis of \(9\) identities that characterize \(3\)-supernilpotent loops. This enables one to classify all \(3\)-supernilpotent loops up to size \(9\).