As a joint work with Almeida and Steinberg, we had continued to work on the pro-nilpotent group topology of a free group. We proved that the pseudovarieties $\mathsf{V} \textcircled{m} \mathsf{G_{nil}}$, for every decidable pseudovariety of monoids $\mathsf{V}$, and $\mathsf{J} \ast \mathsf{G_{nil}}$ are decidable.

At another joint work with Almeida, we had investigated the rank of classes defined by several of the variants of nilpotent semigroups in the sense of Mal'cev. We showed that the pseudovariety $\mathsf{NT}$ has infinite rank and, therefore, it is non-finitely based. We calculated the ranks of these pseudovarieties $\mathsf{MN}$, $\mathsf{\M}$, $\mathsf{PE}$, $\mathsf{TM}$ and $\mathsf{EUNNG}$. They are respectively  $4,3,2,2,$ and $2$. We gave a complete comparison diagram of them. We introduced the pseudovariety $\mathsf{NDF_{12}}$. A finite semigroup $S$ is in \mathsf{NDF_{12}}$ if $S\in\mathsf{BG_{nil}}$, $F_7\not\prec S$ and $F_{12}\not\prec S$.  It is strictly contained in the pseudovariety $\mathsf{PE}$ and strictly includes the pseudovariety $\mathsf{NT}$. On the other hand, $S\in\mathsf{PE}$ if $S\in\mathsf{BG_{nil}}$ and $F_7\not\prec S$. We also proved that $F_7$ is $\times$-prime.

The finite basis property is often connected with the finite rank property. For locally finite varieties and finitely generated pseudovarieties, the two properties are in fact equivalent. As a joint work with Almeida, we started to construct an example which shows that they are not equivalent in the context of pseudovarieties of semigroups.