Nilpotency and strong nilpotency for finite semigroups

Nilpotent semigroups in the sense of Mal'cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, \(\mathsf{MN}\), which has finite rank. The semigroup identities that define nilpotent semigroups, lead us to define strongly Mal'cev nilpotent semigroups. Finite strongly Mal'cev nilpotent semigroups constitute a non-finite rank pseudovariety, \(\mathsf{SMN}\). The pseudovariety \(\mathsf{SMN}\) is strictly contained in the pseudovariety \(\mathsf{MN}\) but all finite nilpotent groups are in \(\mathsf{SMN}\). We show that the pseudovariety \(\mathsf{MN}\) is the intersection of the pseudovariety \(\mathsf{BG_{nil}}\) with a pseudovariety defined by a \(\kappa\)-identity. We further compare the pseudovarieties \(\mathsf{MN}\) and \(\mathsf{SMN}\) with the Mal'cev product of the pseudovarieties \(\mathsf{J}\) and \(\mathsf{G_{nil}}\).