Tesis sobre el tema "Finite semigroups"

A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.

An e-pseudovariety is a class of finite regular semigroups closed under the taking of homomorphic images, regular subsemigroups and finite direct products. Chapter One consists of a survey of those results from algebraic semigroup theory, universal algebra and lattice theory which are used in the following two chapters. In Chapter Two, a theory of generalised existence varieties is developed. A generalised existence variety is a class of regular semigroups closed under the taking of homomorphic images, regular subsemigroups, finite direct products and arbitrary powers. Equivalently, a generalised e-variety is the union of a directed family of existence varieties. It is demonstrated that a class of finite regular semigroups is an e-pseudovariety if and only if the class consists only of the finite members of some generalised existence variety. The relationship between certain lattices of e-pseudovarieties and generalised existence varieties is explored and a usefu l complete surjective lattice homomorphism is found. A study of complete congruences on lattices of existence varieties and e-pseudovarieties forms Chapter Three. In particular it is shown that a certain meet congruence, whose description is relatively simple, can be extended to yield a complete congruence on a lattice of e-pseudovarieties of finite regular semigroups. Ultimately, theorems describing the method of construction of all complete congruences of lattices of e-pseudovarieties whose members are finite E-solid or locally inverse regular semigroups are proved.

In this thesis, we address a number of issues pertaining to the computational power of monoids and semigroups as machines and to the computational complexity of problems whose difficulty is parametrized by an underlying semigroup or monoid and find that these two axes of research are deeply intertwined.
We first consider the "program over monoid" model of D. Barrington and D. Therien [BT88] and set out to answer two fundamental questions: which monoids are rich enough to recognize arbitrary languages via programs of arbitrary length, and which monoids are so weak that any program over them has an equivalent of polynomial length? We find evidence that the two notions are dual and in particular prove that every monoid in DS has exactly one of these two properties. We also prove that for certain "weak" varieties of monoids, programs can only recognize those languages with a "neutral letter" that can be recognized via morphisms over that variety.
We then build an algebraic approach to communication complexity, a field which has been of great importance in the study of small complexity classes. We prove that every monoid has communication complexity O(1), &THgr;(log n) or &THgr;(n) in this model. We obtain similar classifications for the communication complexity of finite monoids in the probabilistic, simultaneous, probabilistic simultaneous and MOD p-counting variants of this two-party model and thus characterize the communication complexity (in a worst-case partition sense) of every regular language in these five models. Furthermore, we study the same questions in the Chandra-Furst-Lipton multiparty extension of the classical communication model and describe the variety of monoids which have bounded 3-party communication complexity and bounded k-party communication complexity for some k. We also show how these bounds can be used to establish computational limitations of programs over certain classes of monoids.
Finally, we consider the computational complexity of testing if an equation or a system of equations over some fixed finite monoid (or semigroup) has a solution.

The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. Although a developed theory of cosets in inverse semigroups exists, it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finte index are much weaker than in the group case. Nevertheless, many aspects of this theory remain of interest, and some of them are addressed in this thesis. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall’s Theorem on counting subgroups of finite index in finitely generated groups. We then look at specific examples, classifying the finite index inverse subsemigroups in polycyclic monoids and in graph inverse semigroups. Finally, we look at the connection between the properties of finite generation and having finte index: these were shown to be equivalent for free inverse monoids by Margolis and Meakin.

In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these.

A semigroup of sets is a family of sets closed under finite unions. This thesis focuses on the search of semigroups of sets in finite dimensional Euclidean spaces Rn, n ≥ 1, which elements do not possess the Baire property, and on the study of their properties. Recall that the family of sets having the Baire property in the real line R, is a σ-algebra of sets, which includes both meager and open subsets of R. However, there are subsets of R which do not belong to the algebra. For example, each classical Vitali set on R does not have the Baire property. It has been shown by Chatyrko that the family of all finite unions of Vitali sets on the real line, as well as its natural extensions by the collection of meager sets, are (invariant under translations of R) semigroups of sets which elements do not possess the Baire property. Using analogues of Vitali sets, when the group 𝒬 of rationals in the Vitali construction is replaced by any countable dense subgroup 𝒬 of reals, (we call the sets Vitali 𝒬-selectors of R) and Chatyrko’s method, we produce semigroups of sets on R related to 𝒬, which consist of sets without the Baire property and which are invariant under translations of R. Furthermore, we study the relationship in the sense of inclusion between the semigroups related to different 𝒬. From here, we define a supersemigroup of sets based on all Vitali selectors of R. The defined supersemigroup also consists of sets without the Baire property and is invariant under translations of R. Then we extend and generalize the results from the real line to the finite-dimensional Euclidean spaces Rn, n ≥ 2, and indicate the difference between the cases n = 1 and n ≥ 2. Additionally, we show how the covering dimension can be used in defining diverse semigroups of sets without the Baire property.

The purpose of this project was to study the classical result in this direction discovered by O. Ore in 1938, as well as related theorems and corollaries. Ore's Theorem and its corollaries provide us with several results relating distributive lattices with cyclic groups.

Les langages réguliers, langages calculés par automates finis, sont parmi les objets les plus simples de l'informatique théorique. Cette thèse étudie plusieurs modèles de calculs: le calcul parallèle avec les circuits booléens, le traitement en flot de documents structurés, et la maintenance d'information sur une structure soumise à des mises à jour incrémentales. Pour ce dernier modèle, les structures auxiliaires sont soit stockées en RAM, soit représentées par des bases de données mises à jour par des formules logiques.Cette thèse étudie les ressources nécessaires pour calculer des classes de langages réguliers dans chacun de ces modèles. Les méthodes employées exploitent l'interaction entre algèbre, logique et combinatoire, en mettant notamment à profit la théorie des semigroupes finis. Cette approche de la complexité s'est notamment montrée extrêmement fructueuse dans le cadre des circuits booléens, où les langages réguliers jouent un rôle central. Cette angle de recherche a été cristallisé par Howard Straubing dans son livre "Finite Automata, Formal Logic, and Circuit Complexity'', où il émet la conjecture que tout langage régulier définissable par une formule arbitraire d'un fragment de logique peut être réécrite en utilisant uniquement des prédicats simples, c'est-à-dire réguliers.Le premier but de ce manuscrit est de prouver cette conjecture dans le cas du fragment Sigma2 de la logique du premier-ordre avec une seule alternance de quantification. Un deuxième résultat propose une description de la complexité en espace, dans le modèle de flot, pour vérifier des propriétés régulières sur des arbres. Une attention particulière est portée aux propriétés vérifiables en espace constant et logarithmique. Un troisième objectif est de décrire tous les langages réguliers d'arbres pouvant être maintenus incrémentalement en temps constant en RAM. Enfin, une dernière partie porte sur le développement de formules logiques efficaces pour maintenir tous les langages réguliers dans le modèle relationnel
Regular languages, languages computed by finite automata, are among the simplest objects in theoretical computer science. This thesis explores several computation models: parallel computing with Boolean circuits, structured document streaming processing, and information maintenance on a structure subject to incremental updates. For the latter, auxiliary structures are either stored in RAM or represented by databases updated by logical formulae.This thesis investigates the resources required to compute classes of regular languages in each of these models. The methods employed rely on the interaction between algebra, logic, and combinatorics, notably exploiting the theory of finite semigroups. This approach of complexity has proven extremely fruitful, particularly in the context of Boolean circuits, where regular languages play a central role. This research angle was crystallised by Howard Straubing in his book "Finite Automata, Formal Logic, and Circuit Complexity", where he conjectured that any regular language definable by an arbitrary formula from a logic fragment can be rewritten to use only simple, regular predicates.The first objective of this manuscript is to prove this conjecture in the case of the Sigma2 fragment of first-order logic with a single alternation of quantification. A second result provides a description of space complexity, in the streaming model, for verifying regular properties on trees. Special attention is given to properties verifiable in constant and logarithmic space. A third objective is to describe all regular tree languages that can be incrementally maintained in constant time in RAM. Finally, a last part focuses on the development of efficient logical formulae for maintaining all regular languages in the relational model

This work presents several results on curves of Hurwitz type, defined over a finite field. In 1961, Tallini investigated plane irreducible curves of minimum degree containing all points of the projective plane PG(2,q) over a finite field of order q. We prove that such curves are Fq3(q2+q+1)-projectively equivalent to the Hurwitz curve of degree q+2, and compute some of itsWeierstrass points. In addition, we prove that when q is prime the curve is ordinary, that is, the p-rank equals the genus of the curve. We also compute the automorphism group of such curve and show that some of the quotient curves, arising from some special cyclic automorphism groups, are still curves of Hurwitz type. Furthermore, we solve the problem of explicitly describing the set of all Weierstrass pure gaps supported by two or three special points on Hurwitz curves. Finally, we use the latter characterization to construct Goppa codes with good parameters, some of which are current records in the Mint table.
Este trabalho apresenta vários resultados em curvas do tipo Hurwitz, definidas sobre um corpo finito. Em 1961, Tallini investigou curvas planas irredutíveis de grau mínimo contendo todos os pontos do plano projetivo PG(2,q) sobre um corpo finito de ordem q. Provamos que tais curvas são Fq3(q2+q+1)-projetivamente equivalentes à curva de Hurwitz de grau q+2, e calculamos alguns de seus pontos de Weierstrass. Em adição, provamos que, quando q é primo, a curva é ordinária, isto é, o p-rank é igual ao gênero da curva. Também calculamos o grupo de automorfismos desta curva e mostramos que algumas das curvas quocientes, construídas a partir de certos grupos cíclicos de automorfismos, são ainda curvas do tipo Hurwitz. Além disso, solucionamos o problema de descrever explicitamente o conjunto de todos os gaps puros de Weierstrass suportados por dois ou três pontos especiais em curvas de Hurwitz. Finalmente, usamos tal caracterização para construir códigos de Goppa com bons parâmetros, sendo alguns deles recordes na tabela Mint.

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe how to use permutation representations of the symmetric group to create and study efficient algorithms that yield such decompositions.

L'objectif principal de cette thèse consiste à étudier plusieurs thématiques : l'étude de l'observation et la commande d'un système de structure flexible et l'étude de la stabilité asymptotique d'un système d'échangeurs thermiques. Ce travail s'inscrit dans le domaine du contrôle des systèmes décrits par des équations aux dérivées partielles (EDP). On s'intéresse au système du corps-poutre en rotation dont la dynamique est physiquement non mesurable. On présente un observateur du type Luenberger de dimension infinie exponentiellement convergent afin d'estimer les variables d'état. L'observateur est valable pour une vitesse angulaire en temps variant autour d'une constante. La vitesse de convergence de l'observateur peut être accélérée en tenant compte d'une seconde étape de conception. La contribution principale de ce travail consiste à construire un simulateur fiable basé sur la méthode des éléments finis. Une étude numérique est effectuée pour le système avec la vitesse angulaire constante ou variante en fonction du temps. L'influence du choix de gain est examinée sur la vitesse de convergence de l'observateur. La robustesse de l'observateur est testée face à la mesure corrompue par du bruit. En mettant en cascade notre observateur et une loi de commande stabilisante par retour d'état, on souhaite obtenir une stabilisation globale du système. Des résultats numériques pertinents permettent de conjecturer la stabilité asymptotique du système en boucle fermée. Dans la seconde partie, l'étude est effectuée sur la stabilité exponentielle des systèmes d'échangeurs thermiques avec diffusion et sans diffusion. On établit la stabilité exponentielle du modèle avec diffusion dans un espace de Banach. Le taux de décroissance optimal du système est calculé pour le modèle avec diffusion. On prouve la stabilité exponentielle dans l'espace Lp pour le modèle sans diffusion. Le taux de décroissance n'est pas encore explicité dans ce dernier cas.

Apresentamos duas construções para unidades de uma ordem em uma classe de álgebras de quatérnios que é anel de divisão: as unidades de Pell e as unidades de Gauss. Classificamos os anéis de inteiros de extensões quadráticas racionais, $R$, cujo grupo de unidades $\\U (R G)$ é hiperbólico para um certo grupo $G$ fixado. Também classificamos os semigrupos finitos $S$, tal que, para a álgebra unitária $\\Q S$ e para toda $\\Z$-ordem $\\Gamma$ de $\\Q S$, o grupo de unidades $\\U (\\Gamma)$ é hiperbólico. Nesse mesmo contexto, classificamos os {\\it RA}-loops $L$ cujo loop de unidades $\\U (\\Z L)$ não contém um subgrupo abeliano livre de posto dois.
For a given division algebra of a quaternion algebra, we construct and define two types of units of its $\\Z$-orders: Pell units and Gauss units. Also, for the quadratic imaginary extensions over the racionals and some fixed group $G$, we classify the algebraic integral rings for which the unit group ring is a hyperbolic group. We also classify the finite semigroups $S$, for which all integral orders $\\Gamma$ of $\\Q S$ have hyperbolic unit group $\\U(\\Gamma)$. We conclude with the classification of the $RA$-loops $L$ for which the unit loop of its integral loop ring does not contain a free abelian subgroup of rank two.

Cette thèse est centrée autour de l'étude théorique et de l'analyse numérique des équations paraboliques non linéaires avec divers conditions aux limites. La première partie est consacrée aux équations paraboliques dégénérées mêlant des phénomènes non-linéaires de diffusion et de transport. Nous définissons des notions de solutions entropiques adaptées pour chacune des conditions aux limites (flux nul, Robin, Dirichlet). La difficulté principale dans l'étude de ces problèmes est due au manque de régularité du flux pariétal pour traiter les termes de bords. Ceci pose un problème pour la preuve d'unicité. Pour y remédier, nous tirons profit du fait que ces résultats de régularités sur le bord sont plus faciles à obtenir pour le problème stationnaire et particulièrement en dimension un d'espace. Ainsi par la méthode de comparaison "fort-faible" nous arrivons à déduire l'unicité avec le choix d'une fonction test non symétrique et en utilisant la théorie des semi-groupes non linéaires. L'existence de solution se démontre en deux étapes, combinant la méthode de régularisation parabolique et les approximations de Galerkin. Nous développons ensuite une approche directe en construisant des solutions approchées par un schéma de volumes finis implicite en temps. Dans les deux cas, on combine les estimations dans les espaces fonctionnels bien choisis avec des arguments de compacité faible ou forte et diverses astuces permettant de passer à la limite dans des termes non linéaires. Notamment, nous introduisons une nouvelle notion de solution appelée solution processus intégrale dont l'objectif, dans le cadre de notre étude, est de pallier à la difficulté de prouver la convergence vers une solution entropique d'un schéma volumes finis pour le problème de flux nul au bord. La deuxième partie de cette thèse traite d'un problème à frontière libre décrivant la propagation d'un front de combustion et l'évolution de la température dans un milieu hétérogène. Il s'agit d'un système d'équations couplées constitué de l'équation de la chaleur bidimensionnelle et d'une équation de type Hamilton-Jacobi. L'objectif de cette partie est de construire un schéma numérique pour ce problème en combinant des discrétisations du type éléments finis avec les différences finies. Ceci nous permet notamment de vérifier la convergence de la solution numérique vers une solution onde pour un temps long. Dans un premier temps, nous nous intéressons à l'étude d'un problème unidimensionnel. Très vite, nous nous heurtons à un problème de stabilité du schéma. Cela est dû au problème de prise en compte de la condition de Neumann au bord. Par une technique de changement d'inconnue et d'approximation nous remédions à ce problème. Ensuite, nous adaptons cette technique pour la résolution du problème bidimensionnel. A l'aide d'un changement de variables, nous obtenons un domaine fixe facile pour la discrétisation. La monotonie du schéma obtenu est prouvée sous une hypothèse supplémentaire de propagation monotone qui exige que la frontière libre se déplace dans les directions d'un cône prescrit à l'avance.

Ce travail est une contribution à l'étude de la limite singulière des équations et des systèmes de Réaction-Diffusion. Ces derniers modélisent des problèmes issus de la physique, de la chimie, de la biologie et des sciences de la technologie. En effet, ce type de problème se présente dans la nature et sont caractérisés par la présence de paramètres qui, lorsqu' ils sont suffisamment grands, donnent lieu généralement à un phénomène appelé couches limites. Cette thèse est composée de cinq chapitres traitant les limites singulières des équations et des systèmes de Réaction Diffusion ainsi que l' existence et l'unicité de solution pour un problème d'obstacle et de quelques EDPs elliptique-parabolique doublement non linéaire avec un opérateur de type Leray Lions. Dans le premier chapitre, nous présentons des résultats théoriques et abstraits sur les limites singulières, où nous traitons aussi la compétition entre deux ou plusieurs opérateurs. Nous appliquons ces résultats dans le contexte des équations aux dérivées partielles et nous étudions le comportement de la solution d'un modèle, lorsque les coefficients de diffusion et/ou de réaction deviennent très grands. Dans les deux chapitres qui suivent, nous considérons un système de réaction diffusion intervenant dans des modèles (macroscopiques) de diffusion dans un milieu hétérogène. Nous présentons d'abord une analyse mathématique (existence et unicité de la solution), ensuite nous étudions le comportement de la solution lorsque le paramètre d'homogénéité devient très grand sur un sous domaine. Le chapitre trois est dédié à l'analyse numérique d'un modèle linéaire, nous prouvons l' existence d'une solution approchée satisfaisant des propriétés de stabilité et de convergence vers la solution du problème continu indépendamment du paramètre d'homogénéité. Le chapitre quatre a pour objet l'étude de l'existence et l'unicité de la solution d'un problème d'obstacle doublement non linéaire avec des contraintes bilatérales, dépendantes de l'espace. Enfin, dans le cinquième chapitre, nous présentons une généralisation des résultats du chapitre trois au cas d'un opérateur de type Leray-Lions et une réaction qui dépend de l'espace.

Dans cette thèse, on se propose d'étudier les automates de Mealy, c'est-à-dire des transducteurs complets déterministes lettre à lettre ayant même alphabet d'entrée et de sortie. Ces automates sont utilisés depuis les années 60 pour engendrer des (semi-)groupes qui ont parfois des propriétés remarquables, permettant ainsi de résoudre plusieurs problèmes ouverts en théorie des (semi-)groupes. Dans ce travail, on s’intéresse plus particulièrement aux apports possibles de l'informatique théorique à l'étude de ces (semi-)groupes engendrés par automate. La thèse présentée s'articule autours de deux grands axes. Le premier, qui correspond aux chapitres II et III, traite des problèmes de décision et plus spécifiquement du problème de Burnside dans le chapitre II et des points singuliers dans le chapitre III. Dans ces deux chapitres on met en lien des propriétés structurelles de l'automate avec des propriétés du groupe engendré ou de son action. Le second axe, représenté par le chapitre IV, se rapporte à la génération aléatoire de groupes finis. On cherche, en tirant des automates de Mealy aléatoirement dans des classes spécifiques, à engendrer des groupes finis, et on aboutit à un résultat de convergence pour la distribution ainsi obtenue. Ce résultat fait écho au théorème de Dixon pour les groupes de permutations aléatoires
In this thesis, we study Mealy automata, i.e. complete, deterministic, letter-to-letter transducers which have same input and output alphabet. These automata have been used since the 60s to generate (semi)groups that sometimes have remarkable properties, that were used to solve several open problems in (semi)group theory. In this work, we focus more specifically on the possible contributions that theoretical computer science can bring to the study of these automaton (semi)groups.The thesis consists of two main axis. The first one, which corresponds to the Chapters II and III, deals with decision problems and more precisely with the Burnside problem in Chapter II and with singular points in Chapter III. In these two chapters, we link structural properties of the automaton with properties of the generated group or of its action. The second axis, which comprises the Chapter IV, is related with random generation of finite groups. We seek, by drawing random Mealy automata in specific classes, to generate finite groups, and obtain a convergence result for the obtained distribution. This result echoes Dixon's theorem on random permutation groups

by Jin Mai.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.
Includes bibliographical references (leaves 74-79).
Acknowledgement --- p.i
Abstract --- p.ii
Chapter 1. --- Pseudovarieties of finite algebras --- p.1
Chapter 2. --- Algebraic automata and formal languages theory --- p.19
Chapter 3. --- M-varieties and S-varieties --- p.36
Chapter 4. --- The dot-depth hierarchy --- p.48
Chapter 5. --- Operators P and P' --- p.62
References --- p.74

In mathematics, one frequently encounters constructions of a pathological or critical nature. In this thesis we investigate such structures in semigroup theory with a particular aim of finding small, finite, examples with certain associated infinite characteristics. We begin our investigation with a study of the identities of finite semigroups. A semigroup (or the variety it generates) whose identities admit a finite basis is said to be finitely based. We find examples of pairs of finite (aperiodic) finitely based semigroups whose direct product is not finitely based (answering a question of M. Sapir) and of pairs of finite (aperiodic) semigroups that are not finitely based whose direct product is finitely based. These and other semigroups from a large class (the class of finite Rees quotients of free monoids) are also shown to generate varieties with a chain of finitely generated supervarieties which alternate between being finitely based and not finitely based. Furthermore it is shown that in a natural sense, "almost all" semigroups from this class are not finitely based. Not finitely based semigroups that are locally finite and have the property that every locally finite variety containing them is also not finitely based are said to be inherently not finitely based. We construct all minimal inherently not finitely based divisors in the class of finite semigroups and establish several results concerning a fundamental example with this property; the six element Brandt semigroup with adjoined identity element, B. 1/2. We then find the first examples of finite semigroups admitting a finite basis of identities but generating a variety with uncountably many subvarieties (indeed with a chain of subvarieties with the same ordering as the real numbers). For some well known classes, a complete description of the members with this property are obtained and related examples and results concerning joins of varieties are also found. A connection between these results and the construction of varieties with decidable word problem but undecidable uniform word problem is investigated. Finally we investigate several embedding problems not directly concerned with semigroup varieties and show that they are undecidable. The first and second of these problems concern the fundamental relations of Green; in addition some small examples are found which exhibit unusual related properties and a problem of M. Sapir is solved. The third of the embedding problems concerns the potential embeddability of finite semigroup amalgams. The results are easily extended to the class of rings.

O objetivo deste trabalho é fornecer um atlas das bases de identidades para as variedades geradas por semigrupos e grupos de ordem pequena. Com o propósito de auxiliar os matemáticos que trabalham neste campo a encontrar informações com facilidade, foi implementado um website que executa em segundo plano um conjunto de algoritmos desenvolvidos no âmbito deste trabalho e ainda ferramentas de demonstração automática e construtores de modelos finitos, para que o utilizador tenha um guia automático e inteligente. Por exemplo, o site fornece a primeira lista completa de variedades geradas por semigrupos de ordens iguais ou inferiores a 5, e consegue identificar rapidamente se a variedade gerada por um semigrupo inserido pelo utilizador, mesmo para ordens bastante superiores, coincide com uma das 218 variedades presentes na base de dados (atualmente inclui ainda outras 11 variedades geradas por semigrupos de ordens superiores). O site também fornece bases de identidades para classes arbitrariamente grandes de semigrupos, como bandas, ou a identificação dos semigrupos de ordem 6 conhecidos que geram variedades de base não finita. Além desta base de dados, o site propõe ainda uma lista das variedades geradas pelos semigrupos de ordem 6, num total de 414 conjeturas para novas variedades, entre as 463 identificadas neste trabalho e retirando as 49 variedades já conhecidas. O site disponibiliza ainda a informação para variedades geradas pelos grupos de ordens iguais ou inferiores a 255, obtida em resultado do levantamento e análise da literatura existente sobre as variedades geradas por estes grupos, para um total de 7012 grupos. O tratamento da informação recolhida para variedades geradas por grupos foi efetuado em GAP e através do desenvolvimento de algoritmos específicos para grupos comutativos e grupos metabelianos, nomeadamente para obtenção das identidades especificas de cada grupo, quando possível. O site oferece ainda um conjunto de funções sobre semigrupos, como encontrar o mínimo lexicográfico das cópias isomórficas de um dado semigrupo. Em relação à propriedade de base inerentemente não finita, o site pode decidir se um determinado semigrupo finito possui ou não essa propriedade; calcular a decomposição do semigrupo em semireticulados; ou converter a sintaxe da tabela de multiplicação de um semigrupo para utilização no GAP ou num demonstrador automático de teoremas/construtor de modelos finitos. O site fornece algumas outras funcionalidades, como uma ferramenta que gera a tabela de multiplicação de um semigrupo fornecida por uma apresentação em C, onde C ´e qualquer classe de álgebras definida por um conjunto de fórmulas em predicados de primeira ordem. A operação no sentido inverso também está acessível. O site disponibiliza um conjunto de informações e funcionalidades sobre variedades e as suas bases de identidades, como apresentar todas as inclusões entre as variedades da base de dados (variedades que são sub-variedades de outras variedades, que as contem) em formato gráfico, identificar se a variedade gerada por um semigrupo coincide com uma existente na base de dados, ou identificar a variedade cuja base é equivalente ao conjunto de identidades inseridas pelo utilizador. O site consegue ainda listar todas as variedades da base dados cujas bases de identidades implicam ou são implicadas por um conjunto de identidades definidas pelo utilizador. Neste cálculo o site leva em conta o conhecimento prévio de todas as inclusões entre variedades acima referidas para acelerar o cálculo e minimizar o uso do demonstrador automático de teoremas/construtor de modelos finitos. No desenvolvimento deste site foram utilizados: na implementação de algoritmos no servidor, Python, substituído pela versão compilada Cython em todos os cálculos intensivos; no desenvolvimento da interface cliente, JavaScript, JQuery, Ajax, Flask, HTML5, CSS3, Bootstrap e MathJax; em bases de dados relacionais, MySQL e SQLAlchemy; na preparação da informação presente nas bases de dados: GAP-System e em particular os “packages” smallsemi e smallgroups; na demonstração automática de teoremas e construção de modelos finitos, Prover9 e Mace4; na apresentação de diagramas, Graphviz. Foi desenvolvido um extenso conjunto de algoritmos reutilizáveis, para manipulação de variedades, semigrupos e grupos, organizados em bibliotecas, destacando-se: varlib.pyx – Implementa os algoritmos de cálculo intensivo do site, como o algoritmo que encontra o mínimo lexicográfico das cópias isomorfas de um dado semigrupo, ou o que verifica se um dado semigrupo satisfaz a base de identidades de uma das variedades na base de dados, e não satisfaz as identidades que definem as sub-variedades maximais. Por questões de velocidade, não existe nesta biblioteca recurso a demonstradores automáticos de teoremas ou a construtores de modelos finitos, sendo a sua funcionalidade substituída por algoritmos desenvolvidos otimizados para identidades, correndo nesta biblioteca a uma velocidade tipicamente 1000x superior ao mesmo programa em Python interpretado; p9m4tools.py – Implementa os algoritmos que recorrem ao demonstrador automático de teoremas e construtor de modelos finitos, embora em ´ultimo recurso, por questões de desempenho, através da implementação de diversas técnicas de “cache”. Nesta biblioteca estão por exemplo os algoritmos desenvolvidos para obter um semigrupo e a sua tabela de multiplicação a partir de uma apresentação, e filtrar as variedades cuja bases de identidades implicam, são implicadas por, ou são equivalentes a um conjunto de identidades entradas pelo utilizador; vartools.py – Implementa rotinas de menor exigência computacional, como a interpretação da entrada de dados do utilizador (por exemplo tabelas de multiplicação em diversos formatos à escolha do utilizador) e a formatação dos dados a apresentar, quer em texto que de forma gráfica. Esta biblioteca inclui ainda diversos algoritmos sobre semigrupos, como a decomposição em semireticulados. Este trabalho também contribui para a extensão da base de dados de variedades conhecidas: existem 28634 semigrupos de ordem 6, considerados equivalentes quando isomorfos ou anti-isomorfos. As variedades de 2035 desses semigrupos não coincidem com nenhuma variedade conhecida gerada por semigrupos de ordem até 5. Neste projeto, com base no teorema de Birkhoff e aplicando novos algoritmos de computador e uma ferramenta de demonstração automática de teoremas/construtor de modelo finitos, foi possível dividir esses 2035 semigrupos em 463 conjuntos de semigrupos que satisfazem as mesmas identidades, correspondendo a 414 novas variedades propostas (dado que são já conhecidas 45 variedades de base finita e 4 variedades de base não finita, geradas por semigrupos da ordem 6). Além disso, as identidades candidatas para a bases de identidades para todas estas novas variedades também são propostas e apresentadas nesta tese, acompanhadas por todas as tabelas de Cayley e os IDs no “package” GAPs smallsemi correspondentes, para cada conjunto de semigrupos encontrados que geram a mesma variedade conjeturada. As provas dessas novas variedades representam problemas em aberto e um desafio para todos os matemáticos nesta área. A mesma metodologia pode ser utilizada para encontrar novas variedades candidatas geradas por semigrupos de ordem 7 ou maior (no âmbito deste trabalho foram encontrados 73807 semigrupos não isomorfos de ordem 7 cujas variedades não coincidem com as variedades registadas na base de dados do site). Esta tese começa com um artigo 1 de pesquisa sobre variedades de semigrupos que contém a maior parte do conteúdo matemático desta tese. Este trabalho representa diversas contribuições para o campo do estudo das variedades de semigrupos: pela primeira vez foi reunida a informação dispersa sobre todas as variedades geradas por semigrupos de ordem igual ou inferior a 5, e relativa aos grupos de ordem igual ou inferior a 255, sendo apresentada num site de fácil utilização. O site oferece ainda um conjunto de funcionalidades sobre semigrupos e sobre variedades da base de dados, alicerçadas num conjunto de algoritmos desenvolvidos para maximização da performance e integrando uma interface, transparente para o utilizador, com um demonstrador automático de teorema e um construtor de modelos finitos, funcionando em paralelo para resultados mais rápidos. Adicionalmente, este trabalho contribuiu para o enriquecimento da base de dados de variedades geradas por semigrupos, ao propor conjeturas para todas as variedades não conhecidas geradas por semigrupos de ordem 6 e respetivas bases de identidades. As principais limitações deste trabalho estão relacionadas com o fato de alguns dos algoritmos desenvolvidos exigirem uma utilização intensiva de recursos computacionais (memória e processador). Estes algoritmos, ou não são adequados para colocação no site (como a pesquisa de novas variedades e respetivas bases), ou exigem limitações dos parâmetros de entrada (por exemplo o cálculo do mínimo lexicográfico, a geração de semigrupos a partir de apresentações, e a identificação de variedades, estão limitados a semigrupos de ordem inferior ou igual a, respetivamente, 10, 20 e 100). Noutras situações, embora raras, o site pode atingir o limite de memória permitido a cada utilizador. Este trabalho inspirou um alargado conjunto de ideias para trabalho futuro, e a primeira consiste na criação de um “package” GAP. Além de oferecer tudo o que site oferece e sem as limitações de memória e processador do site, o utilizador pode realizar múltiplas operações de forma automática. Na realidade o grosso do trabalho está pronto, já que este “package” se pode materializar apenas com o desenvolvimento de uma pequena camada de interface de comandos GAP, já que todos os algoritmos do site estão em bibliotecas autónomas e prontos para ser invocados por qualquer programa externo (aliás o acesso pelo GAP a rotinas das bibliotecas do site foi testado com sucesso no âmbito deste trabalho). Outras ideias para trabalho futuro passam por estender a procura de variedades `as variedades geradas por semigrupos de ordem 7; melhorar os algoritmos atuais para propor também as identidades que definem as sub-variedades maximais; automatizar a demonstração das variedades conjeturadas; desenvolver uma interface no site que permita a extensão da base de dados com novas variedades pelos matemáticos; aumentar as funcionalidades sobre variedades de grupos; desenvolver novo algoritmo para acelerar o cálculo do mínimo lexicográfico.
The aim of this work was to provide an atlas of identity bases for varieties generated by small semigroups and groups. To help the working mathematician easily find information, a website was implemented, running in the background automated reasoning tools and finite model builders, so that the user has an automatic intelligent guide on the literature. For instance, the site provide the first complete the list of varieties generated by a semigroup of order up to 5. The website also provides identity bases for several types of semigroups or groups, such as bands, commutative groups, and metabelian groups. Regarding the inherent non-finite basis property, the website can decide whether or not a given finite semigroup possesses this property. The site provides some other functionalities such as a tool that outputs the multiplication table of a semigroup given by a C-presentation, where C is any class of algebras defined by a set of first order formulas. The inverse conversion is also available. This work also gives a contribution to the extension of the database of known varieties: there are 28634 nonisomorphic semigroups of order 6. The varieties of 2035 of these semigroups do not coincide to any known variety generated by semigroups of order up to 5. In this project, building on the Birkhoff theorem and applying new computer algorithms and automatic theorem proving, it was possible to divide these 2035 semigroups into 463 sets of semigroups that satisfy the same identities, corresponding to 414 new proposed varieties (since there are 45 known finitely-based and 4 known non-finitely based varieties generated by semigroups of order 6). Additionally, candidate identities for the identity basis for all these new varieties are also proposed and presented in this thesis, accompanied by all Cayley tables and the corresponding GAP smallsemi package IDs in each set of semigroups found. The proofs of these new varieties represent open problems and a challenge to all mathematicians in this field. The same methodology can be extended to find new candidate varieties generated by semigroups of order 7 or larger (within this work were found 73807 nonisomorphic semigroups of order 7 whose varieties do not coincide with known varieties registered in the site database).

We investigate the structure of C*-algebras with a finite bound on the dimensions of their irreducible representations, sometimes called 'subhomogenous' In the first chapter we develop the theory of C*-semigroup bundles. These are C*-bundles over semigroups together with a 'structure map' which links the semigroup structure of the base space to the bundle. Under suitable conditions we prove the existence of 'enough' bounded sections, which are 'compatible' with the C*-semigroup bundle structure. Then we establish a complete duality between a certain class of C*-semigroup bundles and subhomogenous C*-algebras, namely the algebra of compatible sections of such a C*-semigroup bundle is subhomogenous and conversely, every subhomogenous C*-algebra is isomorphic to the algebra of compatible sections of such a C*-semigroup bundle. In this way we are able to even represent C*-algebras with non-Hausdorff spectrum as sections in bundles The second chapter is devoted to developing methods for the computation of the functor (PI)H(,R)('1), which classifies certain C*- bundles with varying finite dimensional fibres. (PI)H(,R)('1) is the C*- bundle analog of Cech-cohomology for bundles with one fibre type. The difficulty here is, that homotopy classes of cocycles of bundle imbeddings have to be computed, while only homotopies that satisfy a corresponding cocycle condition can be considered. We define a functor MH(,R)('1) which describes the multiplicities of the imbeddings of the fibres into the bundle and assignment of multiplicity matrices to cocycles yields a natural transformation: (PI)H(,R)('1) (--->) MH(,R)('1) Chapter three finally gives some applications. We calculate (PI)H(,R)('1) for C*-bundles over a two disk for an assignment of different finite dimensional fibres. The result is stated in terms of MH(,R)('1) and quotients of homotopy groups of bundle imbeddings. It provides a new way to describe the group C*-algebra of an interesting group called p4gm, which has been computed by I. Raeburn, and furthermore, our description yields complete invariants--in fact these are given by MH(,R)('1) A last example involving bundles over a three ball with 3 different fibres shows the fact that MH(,R)('1) does not always provide complete invariants and at the same time illustrates the limits of our methods
acase@tulane.edu

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of suffix-, bifix-, and factor-free regular languages, star-free languages including three subclasses, and R- and J-trivial regular languages. We found upper bounds on the syntactic complexities of these classes of languages. For R- and J-trivial regular languages, the upper bounds are n! and ⌊e(n-1)!⌋, respectively, and they are tight for n >= 1. Let C^n_k be the binomial coefficient ``n choose k''. For monotonic languages, the tight upper bound is C^{2n-1}_n. We also found tight upper bounds for partially monotonic and nearly monotonic languages. For the other classes of languages, we found tight upper bounds for languages with small state complexities, and we exhibited languages with maximal known syntactic complexities. We conjecture these lower bounds to be tight upper bounds for these languages. We also observed that, for some subclasses C of regular languages, the upper bound on state complexity of the reversal operation on languages in C can be met by languages in C with maximal syntactic complexity. For R- and J-trivial regular languages, we also determined tight upper bounds on the state complexity of the reversal operation.

Poniższa rozprawa składa się z dwóch części. Obie z nich badają geometryczne własności miar występujących w skończenie wymiarowych układach dynamicznych, głównie z punktu widzenia teorii wymiaru. Część pierwsza dotyczy probabilistycznych aspektów twierdzenia Takensa o zanurzaniu, zajmującego się zagadnieniem rekonstrukcji układu dynamicznego z ciągu pomiarów wykonanych za pomocą jednowymiarowej obserwabli. Klasyczne wyniki z tej dziedziny orzekają, że dla typowej obserwabli, dowolny stan początkowy układu jest jednoznacznie wyznaczony przez ciąg pomiarów, o ile ich ilość przekracza dwukrotnie wymiar przestrzeni fazowej. Główny wynik tej części rozprawy stwierdza, że w kontekście probabilistycznym liczba pomiarów może być dwukrotnie zmniejszona, tzn. prawie każdy stan początkowy układu jest wyznaczony jednoznacznie, o ile ilość pomiarów jest większa od wymiaru Hausdorffa przestrzeni fazowej. Powyższy wynik dowodzi częściowo hipotezy Shroera, Sauera, Otta oraz Yorka z 1998 roku. Przedstawiamy także niedynamiczną wersję probabilistycznego twierdzenia o zanurzaniu oraz szereg przykładów. W drugiej części rozprawy rozważamy rodzinę stacjonarnych miar probabilistycznych dla pewnych losowych układów dynamicznych na odcinku jednostkowym oraz badamy ich własności geometryczne. Rozważane miary mogą być traktowane jako miary stacjonarne dla procesu Markowa na odcinku, otrzymanego przez losowe iterowanie dwóch kawałkami afinicznych homeomorfizmów odcinka. Układy tej postaci nazywamy układami Alsedy-Misiurewicza (albo AM-układami), gdyż badania nad nimi rozpoczęli Alseda oraz Misiurewicz, którzy postawili w 2014 roku hipotezę, że typowe miary stacjonarne dla takich układów są singularne względem miary Lebesgue'a. Głównym celem naszej pracy jest scharakteryzowanie parametrów posiadających tę własność. Naszym głównym wynikiem jest znalezienie pewnych zbiorów parametrów dla których odpowiednie miary są singularne, co dowodzi powyższą hipotezę dla tych zbiorów. Przedstawiamy dwa różne podejścia do dowodzenia singularności - jedno oparte na znajdowaniu minimalnych niezmienniczych zbiorów Cantora oraz drugie, wykorzystujące szacowanie oczekiwanego czasu powrotu do odpowiednio dobranego przedziału. W pierwszym przypadku wyliczamy wymiar Hausdorffa miary stacjonarnej dla pewnych parametrów. Przedstawiamy również kilka dodatkowych wyników dotyczących AM-układów.
This dissertation consists of two parts, both studying geometric properties of measures occuring in finite-dimensional dynamical systems, mainly from the point of view of the dimension theory. The first part concerns probabilistic aspects of the Takens embedding theorem, dealing with the problem of reconstructing a dynamical system from a sequence of measurements performed via a one-dimensional observable. Classical results of that type state that for a typical observable, every initial state of the system is uniquely determined by a sequence of measurements as long as the number of measurements is greater than twice the dimension of the phase space. The main result of this part of the dissertation states that in the probabilistic setting the number of measurements can be reduced by half, i.e. almost every initial state of the system can be uniquely determined provided that the number of measurements is greater than the Hausdorff dimension of the phase space. This result partially proves a conjecture of Shroer, Sauer, Ott and Yorke from 1998. We provide also a non-dynamical probabilistic embedding theorem and several examples. In the second part of the dissertation we consider a family of stationary probability measures for certain random dynamical systems on the unit interval and study their geometric properties. The measures we are interested in can be seen as stationary measures for Markov processes on the unit interval, which arise from random iterations of two piecewise-affine homeomorphisms of the interval. We call such random systems Alseda-Misiurewicz systems (or AM-systems), as they were introduced and studied by Alseda and Misiurewicz, who conjectured in 2014 that typically measures of that type should be singular with respect to the Lebesgue measure. We work towards characterization of parameters exhibiting this property. Our main result is establishing singularity of the corresponding stationary measures for certain sets of parameters, hence confirming the conjecture on these sets. We present two different approaches to proving singularity - one based on constructing invariant minimal Cantor sets and one based on estimating the expected return time to a suitably chosen interval. In the first case we calculate the Hausdorff dimension of the measure for certain parameters. We present also several auxiliary results concerning AM-systems.

W literaturze pojawiło się wiele prac o kratach anihilatorów w algebrach łącznych, oraz o związkach własności tych krat z własnościami krat ideałów jednostronnych i z innymi znanymi, ważnymi własnościami algebr. Celem rozprawy jest kontynuowanie tych badań. Szczególny nacisk jest położony na badanie algebr zredukowanych, oraz algebr półprymarnych, w tym algebr skończenie wymiarowych. Podamy teraz przykłady uzyskanych rezultatów. W rozdziale drugim pokazujemy, że kraty anihilatorów lewostronnych i anihilatorów prawostronnych dowolnej algebry zredukowanej są równe i ta krata jest algebrą Boole'a. Jeśli A jest algebrą półprymarną, to krata anihilatorów tej algebry jest algebrą Boole'a wtedy i tylko wtedy, gdy A jest skończoną sumą prostą algebr z dzieleniem. W rozdziale trzecim, dla dowolnego ciała K i dla dowolnej kraty L konstruujemy lokalną algebrę K[L], taką że jej krata anihilatorów zawiera L jako podkratę. Można przy tym żądać, aby algebra K[L] była przemienna. To pozwala nam udowodnić, że nie istnieje żadna nietrywialna tożsamość spełniona w kratach anihilatorów wszystkich algebr lokalnych. Przypomnijmy, że kraty ideałów jednostronnych we wszystkich algebrach są modularne, a więc spełniają wiele wspólnych, nietrywialnych tożsamości. Jeśli L jest kratą skończoną, to nasza algebra K[L] jest skończenie wymiarowa. Tak więc otrzymujemy przykłady zanurzeń krat skończonych w kraty anihilatorów algebr skończenie wymiarowych, nawet dla tych krat, dla których wcześniej znane były zanurzenia jedynie w kraty anihilatorów algebr nieskończenie wymiarowych. Korzystając ze wspomnianej wyżej konstrukcji opisujemy, w rozdziale czwartym, kraty skończone, które mogą być reprezentowane jako kraty wszystkich anihilatorów algebr skończenie wymiarowych nad ciałami nieskończonymi.
In several papers connections of properties of lattices of annihilators with properties of lattices of one-sided ideals and with other important properties of associative algebras are considered. The aim of this dissertation is to continue these considerations. Special attention will be paid to reduced algebras and semiprimary algebras, in particular to finite dimensional algebras. We provide now some of obtained results. In chapter two we show that for every reduced algebra the lattices of its left annihilators and right annihilators are equal and this lattice is a Boolean algebra. If A is a semiprimary algebra, then the lattices of annihilators of A are Boolean algebras if and only if A is a finite direct sum of division algebras. In the third chapter, for any field K and for arbitrary lattice L we construct a local algebra K[L] and a lattice embedding of L into the lattice of left annihilators of K[L]. In addition we can assume, that K[L] is a commutative algebra. As a consequence, we are able to prove that there is no nontrivial identity satisfied in all lattices of annihilators in local algebras. Let us remind, that lattices of one-sided ideals in all algebras are modular. Thus they satisfy many nontrivial identities. If L is a finite lattice, then our algebra K[L] is finite dimensional. Hence we obtain an embedding of any finite lattice into a lattice of annihilators in a finite dimensional algebra. Earlier examples from the literature showed mainly embeddings into lattices of annihilators of infinite dimensional algebras. Using our construction we also describe, in chapter four, finite lattices being representable as lattices of all annihilators in finite dimensional algebras over infinite fields.

The thesis deals with subsemigroups of (Nm 0 , +), a special discussion is later devoted to the cases m = 1, m = 2 and m = 3. We prove that a subsemigroup of Nm 0 is finitely generated if and only if its generated cone is finitely generated (equivalently polyhedral) and we describe basic topological properties of such cones. We give a few examples illustrating that conditions sufficient for finite generation in N2 0 can not be easily trans- ferred to higher dimensions. We define the Hilbert basis and the related notion of Carathéodory's rank. Besides their basic properties we prove that Carathédory's rank of a subsemigroup of Nm 0 , m = 1, 2, 3, is less than or equal to m. A particular attention is devoted to the subsemigroups containing non-trivial subsemigroups of "subtractive" elements.

Rozprawa ta poświęcona jest badaniu półgrupy C(A) klas sprzężoności ideałów lewostronnych skończenie wymiarowej algebry A z 1 nad ciałem K. Mnożenie w tej półgrupie pochodzi w sposób naturalny od struktury multyplikatywnej samej algebry. Wykazano szereg rezultatów dotyczących struktury C(A). Udowodniono, że skończoność C(A) równoważna jest temu, że liczba klas sprzężoności lewostronnych ideałów nilpotentnych algebry A jest skończona. Wskazano pewne niezmienniki algebry A, które można odczytać ze struktury półgrupy C(A). Przy założeniu, że K jest algebraicznie domknięte oraz, że radykał Jacobsona algebry A jest 2-nilpotentny wykazano, że struktura półgrupy C(A) determinuje strukturę algebry A, przy dodatkowym założeniu, że C(A) jest skończona. W kontekście rozważanej w tym rezultacie klasy algebr, uzyskano także pewne częściowe wyniki związane z opisem algebr, w których półgrupa klas sprzężoności jest skończona. Pokazano, między innymi, że jeśli A jest algebrą z 2-nilpotentnym radykałem Jacobsona, wówczas skończoność półgrupy C(M_6(A)) równoważna jest temu, że A ma skończony typ reprezentacyjny.
The aim of this thesis is to investigate the semigroup C(A) of conjugacy classes of left ideals of a finite dimensional algebra A with 1 over a field K. The operation in this semigroup is naturally induced from the multiplicative structure of the algebra itself. We determine certain invariants of an algebra A that can be expressed in terms of the structure of C(A). Assuming that the field K is algebraically closed and that the Jacobson radical of the algebra A is 2-nilpotent, we prove that the structure of the semigroup C(A) completely determines the structure of A, assuming that C(A) is finite. In the context of the class of algebras that are considered for this result, some partial results concerning the classification of algebras for which the semigroups of conjugacy classes are finite are obtained. It is shown, among other results, that if the algebra A has 2-nilpotent Jacobson radical then the finiteness of C(M_6(A)) is equivalent to the fact, that A is of finite representation type.