Nilpotency and strong nilpotency for finite semigroups

TL;DR

This paper explores the concepts of nilpotency and strong nilpotency in finite semigroups, defining new pseudovarieties and analyzing their relationships and properties within algebraic structures.

Contribution

It introduces the pseudovariety $ extsf{SMN}$ of strongly Mal'cev nilpotent semigroups and compares it with the classical $ extsf{MN}$, establishing their structural differences and intersections.

Findings

$ extsf{SMN}$ is a non-finite rank pseudovariety contained in $ extsf{MN}$.

$ extsf{MN}$ is the intersection of $ extsf{BG}_{nil}$ and a $ ext{kappa}$-identity defined pseudovariety.

Finite nilpotent groups are included in $ extsf{SMN}$.

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 $\mathsf{G_{nil}}$.