Nilpotent semigroups in the sense of Mal'cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, 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, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BG_{nil} with a pseudovariety defined by a k-identity. We further compare the pseudovarieties MN and SMN with the Mal'cev product of the pseudovarieties J and G_{nil}.
(Joint work with J. Almeida from FCUP-CMUP and M. Kufleitner from FMI, University of Stuttgart)