The presentation explores the idea of introducing the notion of a total order to regular languages and deterministic finite automata (DFAs) with a focus on the definitions as given in a paper by Shyr and Thierrin in 1974. We discuss some properties and consequences of these definitions and present a selection of known results mainly related to state complexity. Finally, we outline a series of interesting open problems that are going to be subject to further investigation.

A Kirkman Triple System (KTS) is called $m$-pyramidal if there exists a subgroup $G$ of its automorphism group that fixes $m$ points of the KTS and acts regularly on the other points. Such a group $G$ admits a unique conjugacy class $C$ of involutions (elements of order 2) and $|C|=m$. We call groups with this property $m$-pyramidal. We prove that, if $m$ is an odd prime power $p^k$, then every $m$-pyramidal group is solvable if and only if either $m=9$ or $k$ is odd. We also determine the sizes of the vertex sets of the $m$-pyramidal KTS when $m$ is a prime number.

This presentation will focus on extending the theory of formations of groups to inverse semigroups. The definitions and properties established for finite groups will serve as a starting point, from which we will define formations of finite inverse semigroups. Furthermore, we will examine wider classes, including i-formations (where i represents idempotent-separating) and some classes named f-formations (where f stands for fundamental).

We consider germs of holomorphic vector fields at the origin in dimension 3 with non-isolated singularities that are tangent to a holomorphic foliation of codimension one, referred to as a 2-flag of foliations. Our focus lies on cases where this geometric structure originates from second-orde

We show that quenched limiting hitting distributions are
compound Poisson distributed for certain random dynamical systems with
targets.
Targets are random and assumed to have well-defined return statistics
of certain type, which turn out to characterize the said compound
Poissonian limit.
Moreover, quenched and annealed polynomial decay of correlations are
assumed, whereas annealed Kac-time normalization is adopted.
Examples discussed are one-dimensional random piecewise expanding systems.

Seguindo a ideia no artigo de Hu e Vercruysse [1], introduzimos morfismos parciais em uma categoria arbitrária $\textbf{C}$, de modo que ações parciais de um monoide $M$ em um conjunto $X$ correspondem a certas funções de $M$ para o conjunto de classes de isomorfismo de morfismos parciais de $X$ para $X$ na categoria de conjuntos.

In this talk we present some properties of generalized torsion elements in groups. Moreover, we try connect this "new'' concept with the usual concept of torsion in some standard class of groups (e.g., nilpotent, FC-groups). This presentation is mainly based in the following papers [1,2,3,4]. This is joint work with C. Schneider and D. Silveira.   
      

References

[1] R. Bastos, C. Schneider and D. Silveira. Generalized torsion elements in groups.  To appear in Arch. Math. Basel (2023), arXiv:2302.09589.  

Let us consider the action of the general linear group $\mathrm{GL}_n(\mathbb{C})$ on the direct product $\mathcal{M}_n^d$
of $d$ copies of $\mathcal{M}_n$ by simultaneous conjugation sending $(X_1,\ldots, X_d)$ to $(gX_1g^{-1},\ldots,gX_dg^{-1})$
for any $g\in \mathrm{GL}_n(\mathbb{C})$ . This induces an action of $\mathrm{GL}_n(\mathbb{C})$ on the algebra $\mathbb{C}[\mathcal{M}_n^d]$ of polynomial
functions on $\mathcal{M}_n^d$. The algebra of invariants under this action, $\mathbb{C}[\mathcal{M}_n^d]^{\mathrm{GL}_n}$, is an important

Motivated by the recent surge of interest in the concept of injective C*-algebras, we return to an old result of Gleason characterizing the projective cover of a given compact topological space.

We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. 

The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral at the origin. 

We propose a new  algorithm to obtain a generating set for the Sibirsky ideal of such polynomial  systems and investigate  some algebraic properties of this ideal.