Nilpotency and strong nilpotency for finite semigroups

Abstract

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 $\mathsf{G_{nil}}$.