Abstract
We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.
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