Let K be a field and Q a finite directed multi-graph. In this paper I classify all path algebras KQ and admissible orders with the property that all of their finitely generated ideals have finite Groebner bases. / Master of Science

A tecnologia de virtualização de redes é um novo paradigma de redes que permite a múltiplas redes virtuais (VNs) compartilharem de uma forma eficiente e eficaz a mesma rede de infraestrutura denominada rede de substrato (SN). A implementação e o desenvolvimento de novos protocolos, testes de novas soluções e arquiteturas para a Internet atual e do futuro podem ser tratadas por meio da virtualização de redes. Com a virtualização de redes surge um desafio denominado problema VNE. O problema de virtualização de redes embutidas (VNE) consiste em realizar o mapeamento dos nós virtuais e o mapeamento dos enlaces virtuais sobre uma rede de substrato (SN). O problema é conhecido como NP-Hard e a sua solução é realizada por meio de algoritmos heurísticos e aproximados que realizam o mapeamento de nós e enlaces virtuais em dois estágios de forma independente ou coordenada. A presente tese tem o objetivo de resolver o mapeamento dos enlaces virtuais do problema VNE com a utilização da álgebra de caminhos. A solução apresentada fornece o melhor desempenho quando comparada com as demais soluções de virtualização de redes encontradas na literatura. Os resultados obtidos nas simulações para o problema VNE foram avaliados e analisados com a utilização do algoritmo desenvolvido nesta tese denominado Path Algebra for Virtual Link Mapping (PAViLiM), que utiliza a álgebra de caminhos para realizar o mapeamento de enlaces virtuais para caminhos na rede de substrato. A álgebra de caminhos é poderosa e flexível. Tal flexibilidade permite que ocorra uma exploração detalhada do espaço de soluções e a identificação do melhor critério e política que devem ser utilizados para a virtualização de redes. / The network virtualization technology is a new paradigm of network that allows multiple virtual networks (VNs) share in an efficient and effective way the same network infrastructure called substrate network (SN). The implementation and the development of new protocols, testing of new solutions and architectures for current and future Internet can be addressed through network virtualization. With the network virtualization arises a challenge called VNE problem. The problem of virtual network embedded (VNE) is to conduct the mapping of the virtual nodes and mapping of the virtual links over a substrate network (SN).The problem is known as NP-Hard and its solution is accomplished by means of approximate and heuristic algorithms that perform the mapping of virtual nodes and links in two stages independently or coordinated. This thesis aims to solve the mapping of virtual links for VNE problem using the paths algebra. The solution presented provides the best performance when compared with other networks virtualization solutions from the literature. The results of simulation for the VNE problem were evaluated and analyzed using the algorithm developed in this thesis called Path Algebra for Virtual Link Mapping (PAViLiM), which uses the paths algebra to perform the mapping of virtual links to paths in substrate network. The paths algebra is powerful and flexible. This flexibility allows the occurrence of a detailed exploration for identifying the best solutions and political criteria to be used for network virtualization.

A tecnologia de virtualização de redes é um novo paradigma de redes que permite a múltiplas redes virtuais (VNs) compartilharem de uma forma eficiente e eficaz a mesma rede de infraestrutura denominada rede de substrato (SN). A implementação e o desenvolvimento de novos protocolos, testes de novas soluções e arquiteturas para a Internet atual e do futuro podem ser tratadas por meio da virtualização de redes. Com a virtualização de redes surge um desafio denominado problema VNE. O problema de virtualização de redes embutidas (VNE) consiste em realizar o mapeamento dos nós virtuais e o mapeamento dos enlaces virtuais sobre uma rede de substrato (SN). O problema é conhecido como NP-Hard e a sua solução é realizada por meio de algoritmos heurísticos e aproximados que realizam o mapeamento de nós e enlaces virtuais em dois estágios de forma independente ou coordenada. A presente tese tem o objetivo de resolver o mapeamento dos enlaces virtuais do problema VNE com a utilização da álgebra de caminhos. A solução apresentada fornece o melhor desempenho quando comparada com as demais soluções de virtualização de redes encontradas na literatura. Os resultados obtidos nas simulações para o problema VNE foram avaliados e analisados com a utilização do algoritmo desenvolvido nesta tese denominado Path Algebra for Virtual Link Mapping (PAViLiM), que utiliza a álgebra de caminhos para realizar o mapeamento de enlaces virtuais para caminhos na rede de substrato. A álgebra de caminhos é poderosa e flexível. Tal flexibilidade permite que ocorra uma exploração detalhada do espaço de soluções e a identificação do melhor critério e política que devem ser utilizados para a virtualização de redes. / The network virtualization technology is a new paradigm of network that allows multiple virtual networks (VNs) share in an efficient and effective way the same network infrastructure called substrate network (SN). The implementation and the development of new protocols, testing of new solutions and architectures for current and future Internet can be addressed through network virtualization. With the network virtualization arises a challenge called VNE problem. The problem of virtual network embedded (VNE) is to conduct the mapping of the virtual nodes and mapping of the virtual links over a substrate network (SN).The problem is known as NP-Hard and its solution is accomplished by means of approximate and heuristic algorithms that perform the mapping of virtual nodes and links in two stages independently or coordinated. This thesis aims to solve the mapping of virtual links for VNE problem using the paths algebra. The solution presented provides the best performance when compared with other networks virtualization solutions from the literature. The results of simulation for the VNE problem were evaluated and analyzed using the algorithm developed in this thesis called Path Algebra for Virtual Link Mapping (PAViLiM), which uses the paths algebra to perform the mapping of virtual links to paths in substrate network. The paths algebra is powerful and flexible. This flexibility allows the occurrence of a detailed exploration for identifying the best solutions and political criteria to be used for network virtualization.

The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.  In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Quasi-hereditary algebras are an important class of algebras with many appli-cations in representation theory, most notably the representation theory of semi-simple complex Lie-algebras. Such algebras sometimes admit an exact Borel sub-algebra, that is a subalgebra satisfying similar formal properties to the Borel sub-algebras from Lie theory. This thesis is divided into two parts. In the first part we classify quasi-hereditary algebras with two simple modules over perfect fields up to Morita equivalence, generalizing a similar result by Membrillo-Hernandez for thealgebraically closed case. In the second part, we take a poset X, a certain set M of constants, and a finite set ρ of paths in the Hasse-diagram of X and construct analgebra A(X, M, ρ) that generalizes the twisted double incidence algebras originally introduced by Deng and Xi. We provide necessary and sufficient conditions for this algebra to be quasi-hereditary when X is a tree, and we show that A(X, M, ρ) admits an exact Borel subalgebra when these conditions are satisfied. Following this, we compute the Ext-algebra of the standard modules of A(X, M, ρ).