Semigroups, Automata and Languages

Orderable pseudovarieties

Speaker: 

Jorge Almeida

Date: 

Friday, 15 June, 2018 - 14:30

Venue: 

Room FC1 030, DMat-FCUP

Pseudovarieties of ordered semigroups have been introduced as a refined algebraic classifying tool for regular languages in the sense of Eilenberg's Correspondence Theorem. Forgetting the order, each such pseudovariety generates a pseudovariety of semigroups. A natural question is which pseudovarieties of semigroups arise in this way from non-selfdual pseudovarieties of ordered semigroups which, for the purpose of the talk, we call orderable.

Local finiteness for Green relations in varieties of semigroups.

Speaker: 

Filipa Soares de Almeida

Date: 

Thursday, 26 April, 2018 - 14:30

Venue: 

Room FC1.029

In this talk, we explore a notion that sits between the concept of locally finite variety and that of periodic variety, using the inescapable Green's relations. Namely, a variety is said to be K-finite, where K stands for any of the Green's relations, if every finitely generated semigroup in this variety has but finitely many K-classes. Our characterization uses the language of "forbidden objects".

A graph approach to the structure of locally inverse semigroups.

Speaker: 

Luís Oliveira

Date: 

Friday, 20 April, 2018 - 14:30

Venue: 

Room FC1.030

Let X be a set, X' be a disjoint copy of X and $\bar{X}\wedge\bar{X}=\{(x\wedge y): x,y\in X\cup X'\}$. We look at $\hat{X}=X\cup X'\cup(\bar{X}\wedge \bar{X})$ as a set of letters and consider the free semigroup $\hat{X}^+$ on the set $\hat{X}$. Auinger [1] constructed a model for the bifree locally inverse semigroup on X as a quotient semigroup of $\hat{X}^+$. This result enables us to talk about presentations $\langle X;R\rangle$ of locally inverse semigroups (LI-presentations) where $R\subseteq \hat{X}^+\times\hat{X}^+$.

Pro-p groups, amalgamations, and trees.

Speaker: 

​Theo Zapata

Date: 

Friday, 23 March, 2018 - 14:30

Venue: 

Room FC1.030, DMat-FCUP

We present the result that, under a certain condition, free pro-$p$ products with procyclic amalgamation inherit from its free factors the property of each 2-generator pro-$p$ subgroup being free pro-$p$. This generalizes known pro-$p$ results, as well as some pro-$p$ analogues of classical results in Combinatorial Group Theory.

On the semigroup rank of a group.

Speaker: 

Mário Branco

Date: 

Friday, 16 March, 2018 (All day)

Venue: 

Room FC1.030, DMat-FCUP at 14:30

For an arbitrary group G, it is shown that either the semigroup rank GrkS equals the group rank GrkG, or GrkS = GrkG +1. This is the starting point for the rest of the work, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of any relatively free group is computed. For a finitely generated abelian group G, it is proven that GrkS = GrkG+1 if and only if G is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case.

On the problem of counting numerical semigroups by genus.

Speaker: 

Manuel Delgado

Date: 

Friday, 2 February, 2018 (All day)

Venue: 

Room FC1.030, DMat-FCUP at 14:30

A numerical semigroup is a submonoid of the non-negative integers, under addition, whose complement in IN is finite. The cardinality of this complement is said to be the genus of the numerical semigroup. In 2008 Bras-Amorós conjectured that the sequence $(n_g)_g$, where $n_g$ is the number of numerical semigroups of genus $g$, behaves like the Fibonacci sequence.

On the Dowling and Rhodes lattices and wreath products.

Speaker: 

Pedro V. Silva

Date: 

Friday, 19 January, 2018 (All day)

Venue: 

Room FC1 030, DMat-FCUP at 14:30

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of Boolean representable simplicial complexes. This turns out to be the direct sum of a complete matroid with a lift matroid of the complete biased graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar biased graph.

Rational Embeddings of Hyperbolic Groups.

Speaker: 

Francesco Matucci

Date: 

Friday, 5 January, 2018 (All day)

Venue: 

Room FC1 030, DMat-FCUP at 14:30

For a finitely generated group, the Cayley graph is a metric space encoding the structure of the group. Gromov introduced the notion of a $\delta$-hyperbolic group, a finitely generated group with a negatively curved Cayley graph, that is, for any triangle in the graph with geodesic sides, each side is contained in the $\delta$-neighborhood of the union of the two other sides. Hyperbolic groups are "prevalent" among finitely generated groups.

Automaton (semi)groups: on the undecidability of some problems.

Speaker: 

Emanuele Rodaro

Date: 

Friday, 15 December, 2017 (All day)

Venue: 

Room FC1 004, DMat-FCUP at 14:30

We consider algorithmic problems for automaton semigroups and automaton groups of the freeness and finiteness kind. We first show that checking whether an automaton group has empty set of positive relations is undecidable. Moreover we prove that the emptyness of the set of positive relations is equivalent to the dynamical property of having all the orbital graphs centred at the non-singular points which are acyclic. We also settle the problem of checking the freeness for the semigroup defined by an automaton group by proving that such problem is undecidable.

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