Languages associated with saturated formations of groups

In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The first author acknowledges support from MEC. The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day

Pseudovariedades de grupos e variedades de linguagens associadas

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2008O principal objectivo deste trabalho consiste em dar uma descrição das linguagens reconhecidas pelos grupos super-resolúveis finitos. Essa descrição será feita de dois modos distintos: através de produtos modulares concatenados, mostrando que uma tal linguagem pertence à álgebra de Boole gerada por produtos modulares concatenados de linguagens comutativas elementares e, através de transdutores, provando que essas linguagens são combinações Booleanas de linguagens da forma rτ−1, em que p é um número primo, r ∈ Zp e τ : A∗ → Zp é uma função realizada por algum transdutor na forma triangular estrita.Com vista a esse estudo, faremos uma análise detalhada da pseudovariedade dos grupos super-resolúveis e também de outras pseudovariedades de grupos, em particular, das pseudovariedades dos p-grupos e dos grupos abelianos cujo expoente divide um dado natural n. Caracterizaremos também o produto de pseudo variedades e daremos especial atenção à pseudovariedade Gp ∗ Abp−1. Estudaremos as variedades de linguagens associadas às pseudovariedades de grupos consideradas e iremos demonstrar o Princípio do Produto em Coroa de Straubing, o qual nos fornece uma descrição das linguagens reconhecidas pelo produto em coroa de dois monóides. Além disso, apresentaremos uma versão deste princípio para variedades de linguagens. Será ainda considerado o produto de linguagens com contador e descrita a operação entre monóides que lhe está associada.The main subject of this work is to give a description of the languages recognized by finite super-soluble groups. That description will be done in two distinct ways. The first one uses the modular concatenation product, more precisely, we will prove that such a language is in the Boolean algebra generated by the concatenated modular products of elementary commutative languages. In the second one we prove that the languages recognized by super-soluble groups are Boolean combinations of languages that take the form of rτ−1, where p is a prime number, r ∈ Zp and τ : A∗ → Zp is a function realized by some transductor in the strict triangular form. In view of that study, we will analyse in detail the pseudovarieties of super-soluble groups as well as other pseudovarieties of groups, in particular we will consider the pseudovariety of p-groups and the pseudovariety of abelian groups whose exponent divides a given natural n. We will also characterize the product of pseudovarieties, dedicating particular attention to the pseudovariety Gp ∗ Abp−1.We will study the varieties of languages associated with the pseudovarieties of groups considered and will prove the Straubing's Wreath Product Principle, which gives us a description of the languages recognized by the wreath product of two monoids. In addition, we will present a version of this principle applied to varieties of languages. The product of languages with counter will also be considered and the associated operation between monoids will be described

Languages recognized by finite supersoluble groups

In this paper, we give two descriptions of the languages recognized by finite supersoluble groups. We first show that such a language belongs to the Boolean algebra generated by the modular products of elementary commutative languages. An elementary commutative language is defined by a condition specifying the number of occurrences of each letter in its words, modulo some fixed integer. Our second characterization makes use of counting functions computed by transducers in strict triangular form. Eilenberg’s variety theorem [4] is a powerful tool for classifying regular languages. It states that, given a variety of finite monoids V, the class of languages V whose syntactic monoid belongs to V is a variety of languages, that is, a class of regular languages closed under finite union, complement, left and right quotients and inverse of morphisms. Further, the correspondence V → V between varieties of finite monoids and varieties of languages is one-to-one and onto. Eilenberg’s theorem can be used in both ways: given a variety of languages

Сборник научных работ студентов Республики Беларусь «НИРС 2012» / редкол.: А. И. Жук (пред.) [и др.]

Сборник включает статьи лауреатов, а также авторов работ первой категории XIX Республиканского конкурса научных работ студентов 2012 года. Статьи рекомендованы к опубликованию редакционной коллегией и печатаются в виде, предоставленном авторами, без дополнительного редактирования