Mal'cev algebras with supernilpotent centralizers
Let \({\mathcal A}=(A,F)\) be a finite algebra in a congruence permutable variety. A congruence \(\Theta\) on \({\mathcal A}\) is called \(n\)-supernilpotent if the \((n+1)\)-ary commutator \([\Theta,\ldots,\Theta]\) is the equality relation on \(A\). If for every subdirectly irreducible homomorphic image of \({\mathcal A}\) the centralizer of the monolith is \(n\)-supernilpotent then the clone of polynomial functions on \({\mathcal A}\) is determined by relations of arity \(|A|^{n+1}\). By this result, the author obtains finite implicit descriptions of the polynomial functions on finite local rings with \(1\) and on finite groups \({\mathcal G}\) such that in every subdirectly irreducible quotient of \({\mathcal G}\) the centralizer of the monolith is a \(p\)-group.
scientific article; zbMATH DE number 5886757
scientific article