We investigate the rank of pseudovarieties defined by several variants nilpotency conditions for semigroups in sense Mal'cev. For them, we provide finite bases pseudoidentities. also show that Neumann-Taylor variant does not have rank.

In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.

A mixed matrix pencil is a matrix pencil which has two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. In this paper, we characterize the indices of nilpotency of the Kronecker canonical form for a mixed matrix pencil in terms of matroids. As a byproduct, we provide an algorithm for computing t...

Extending the work of Fröhlich, Lue and Furtado-Coelho, we consider the theory of Baer invariants in the context of semi-abelian categories. Several exact sequences, relative to a subfunctor of the identity functor, are obtained. We consider a notion of commutator which, in the case of abelianization, corresponds to Smith’s. The resulting notion of centrality fits into Janelidze and Kelly’s the...

We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of thes...