As a joint work with Almeida and Steinberg, we had continued to work on the pro-nilpotent group topology of a free group. We studied the pro-nilpotent group topology on a free group. First, we described the closure of the product of finitely many finitely generated subgroups of a free group in the pro-nilpotent group topology and then presented an algorithm to compute it. We deduced that the nil-closure of a rational subset of a free group is an effectively constructible rational subset and hence has a decidable membership.  We also proved that the $\mathsf{G_{nil}}$-kernel of a finite monoid is computable and hence pseudovarieties of the form $\mathsf{V} \smalcev \mathsf{G_{nil}}$ have decidable membership problem, for every decidable pseudovariety of monoids $\mathsf{V}$.  Finally, we proved that the semidirect product $\mathsf{J} \ast \mathsf{G_{nil}}$ has a decidable membership problem. This work is published in the international journal Journal of Algebra.

Mal'cev and independently Neumann and Taylor have shown that nilpotent groups can be defined by semigroup identities (that is, without using inverses). This leads to the notion of a nilpotent semigroup (in the sense of Mal'cev). For a semigroup $S$ with elements $x,y,z_{1},z_{2}, \ldots$ one recursively defines two sequences $$\lambda_n=\lambda_{n}(x,y,z_{1},\ldots, z_{n})\quad{\rm and} \quad \rho_n=\rho_{n}(x,y,z_{1},\ldots, z_{n})$$ by $$\lambda_{0}=x, \quad \rho_{0}=y$$ and $$\lambda_{n+1}=\lambda_{n} z_{n+1} \rho_{n}, \quad \rho_{n+1}=\rho_{n} z_{n+1} \lambda_{n}.$$ A semigroup $S$ is said to be \emph{nilpotent} if there exists a positive integer $n$ such that $$\lambda_{n}(x,y,z_{1},\ldots, z_{n}) = \rho_{n}(x,y,z_{1},\ldots, z_{n})$$ for all $x,y$ in $S$ and $z_{1}, \ldots, z_{n}$ in $S^{1}$. The finite nilpotent semigroups constitute a pseudovariety which is denoted by $\mathsf{MN}$. As a joint work with my Almeida and Kufleitner, we introduced a further variant of Mal'cev nilpotency, that we call strong Mal'cev nilpotency. For semigroups, the new notion is strictly stronger than Mal'cev nilpotency, but it coincides with nilpotency for groups. Strongly Mal'cev nilpotent semigroups constitute a pseudovariety which we denote by $\mathsf{SMN}$. We showed that $\mathsf{G_{nil}}\subsetneqq\mathsf{SMN}\subsetneqq \mathsf{MN}$. Higgins and Margolis showed that $\langle\mathsf{A}\cap\mathsf{Inv}\rangle\subsetneqq \mathsf{A}\cap\langle\mathsf{Inv}\rangle$ \cite{Hig-Mar}. We showed that $\langle\mathsf{A}\cap\mathsf{Inv}\rangle \subsetneqq\mathsf{A}\cap\mathsf{SMN}$. Note that the following chain of proper inclusions holds: $$\mathsf{G_{nil}}\subsetneqq\mathsf{SMN}\subsetneqq \mathsf{MN}\subsetneqq\mathsf{BG}_{nil}.$$ On the other hand, it is well known that $\mathsf{BG}=\mathsf{J} \malcev \mathsf{G}$ where $\malcev$ stands for Mal'cev product. In contrast, the inclusion $\mathsf{J} \malcev \mathsf{H}\subsetneqq \mathsf{BH}$ is strict for every proper subpseudovariety $\mathsf{H}$ of $\mathsf{G}$ \cite{Hig-Mar}. This work has been submitted to an international journal.

The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based. As a joint work with Almeida, we established a result on nilpotent pseudovarieties which yields many similar examples in the realm of pseudovarieties of semigroups. This work is published in the international journal Semigroup Forum.

It is proved that the pseudovariety $\mathsf{J}\ast \mathsf{G_{nil}}$ is decidable by exhibiting an algorithm that solves its membership problem. In joint work with Almeida, we compared the pseudovarieties $\mathsf{MN}$ and $\mathsf{SMN}$ with the pseudovariety $\mathsf{J} \ast \mathsf{G_{nil}}$. We further give examples of Sch\"utzenberger products of monoids which are nilpotent and examples which are not nilpotent. We investigated the nilpotency and strong Mal'cev nilpotency of Sch\"utzenberger products of monoids. Even with very simple monoids as factors, such as cyclic groups, the Sch\"utzenberger product may not be nilpotent. This work is in progress.

Higgins and Margolis stated that $\mathsf{V} \malcev \mathsf{G}= \mathsf{EV}$ holds for the pseudovariety of all finite groups for any $\mathsf{V}$ such that $\mathsf{Sl}\subseteq \mathsf{V} \subseteq \mathsf{DA}$. Almeida has constructed an unpublished counterexample to this statement. As joint work, we have started to characterize the class of such examples. This work is in progress.