# M. Hossein Shahzamanian's Annual Report

2015

## Brief description of the research activities:

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$-prime. The finite basis property is often connected with the finite rank property. For locally finite varieties and finitely generated pseudovarieties, the two properties are in fact equivalent. As a joint work with Almeida, we started to construct an example which shows that they are not equivalent in the context of pseudovarieties of semigroups. ## Seminars and courses given at CMUP or conferences: ## Talks / Seminars / Courses : ### Communications in international conferences Title: The pro-nilpotent group topology on a free group. Name of the event: The Joint AMS-EMS-SPM International Meeting Speakers M. Hossein Shahzamanian Talk: Contributed talk Date: 10.06.2015 to 13.06.2015 Country: Portugal Location / City: Porto Title: The congruence$\eta^{\ast}\$ on semigroups.
Name of the event:
Advances in Group Theory and Applications conference
Speakers
M. Hossein Shahzamanian
Talk:
Contributed talk
Date:
16.06.2015 to 19.06.2015
Country:
Italy
Location / City:
Porto Cesareo, Lecce

### Seminars

Title:
The rank of variants of nilpotent pseudovarieties.
Speakers
M. Hossein Shahzamanian
Date:
27.11.2015
Host institution:
FMI
Country:
Germany
Location / City:
Stuttgart
Title:
The rank of variants of nilpotent pseudovarieties.
Speakers
M. Hossein Shahzamanian
Date:
15.12.2015
Host institution:
CMUC
Country:
Portugal
Location / City:
Coimbra

## Prizes / Distinctions:

I participated at confrence AutoMathA 2015, Jewels of Automata: from Mathematics to Applications, May 2015, Leipzig, Germany. My travel expenses were supported by the conference. http://www.automatha.uni-leipzig.de/

## Work visits:

I am a member of the trilateral project for the years 2015 and 2016 involving Porto University (Portugal), Stuttgart University (Germany) and Bordeaux University (France). The people in charge at each country are, respectively, Jorge Almeida (Porto), Manfred Kueitner (Stuttgart) and Marc Zeitoun (Bordeaux). I have been at FMI, Stuttgart University, Germany, for 23-27 November 2015, as a visiting researcher.