Title: The realization problem for von Neumann regular rings Author: Samuel Mokriš Department: Department of Algebra Supervisor of the master thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: With every unital ring R, one can associate the abelian monoid V (R) of isomor- phism classes of finitely generated projective right R-modules. Said monoid is a conical monoid with order-unit. Moreover, for von Neumann regular rings, it satisfies the Riesz refinement property. In the thesis, we deal with the question, under what conditions an abelian conical re- finement monoid with order-unit can be realized as V (R) for some unital von Neumann regular ring or algebra, with emphasis on countable monoids. Two generalizations of the construction of V (R) to the context of nonunital rings are presented and their interrelation is analyzed. To that end, necessary properties of rings with local units and modules over such rings are devel- oped. Further, the construction of Leavitt path algebras over quivers is presented, as well as the construction of a monoid associated with a quiver that is isomorphic to V (R) of the Leavitt path algebra over the same quiver. These methods are then used to realize directed unions of finitely generated free abelian monoids as V (R) of algebras over any given field. A method...

In this thesis are shown some results of realization of refinement monoid by Grothendieck monoids of von Neumann regular ring. The most important result of this thesis, recently published by P. Ara, J. Bosa and E. Pardo, claims that every finitely generated conical refinement monoid is realizable. We construct for each such monoid a von Neumann regular algebra such that monoid is realizable by this algebra. For this construction we use adaptable separated graphs and their relation to refinement monoids. 1

Stuttgart, Univ., Diss., 2006.

University, Diss., 1997--Saarbrücken.

The Generalised Star-Height Problem is an open question in the field of formal language theory that concerns a measure of complexity on the class of regular languages; specifically, it asks whether or not there exists an algorithm to determine the generalised star-height of a given regular language. Rather surprisingly, it is not yet known whether there exists a regular language of generalised star-height greater than one. Motivated by a theorem of Thérien, we first take a combinatorial approach to the problem and consider the languages in which every word features a fixed contiguous subword an exact number of times. We show that these languages are all of generalised star-height zero. Similarly, we consider the languages in which every word features a fixed contiguous subword a prescribed number of times modulo a fixed number and show that these languages are all of generalised star-height at most one. Using these combinatorial results, we initiate work on identifying the generalised star-height of the languages that are recognised by finite semigroups. To do this, we establish the generalised star-height of languages recognised by Rees zero-matrix semigroups over nilpotent groups of classes zero and one before considering Rees zero-matrix semigroups over monogenic semigroups. Finally, we explore the generalised star-height of languages recognised by finite groups of a given order. We do this through the use of finite state automata and 'count arrows' to examine semidirect products of the form A x Zr where A is an abelian group and Zr is the cyclic group of order r.

No description available.

Nous étudions un certain monoïde défini par une présentation, notée P, qui est un hybride de celles du monoïde de tresses infinies et du monoïde de Thompson. Pour cela, nous utilisons plusieurs approches. On décrit d’abord un système de réécriture convergent pour la présentation P, ce qui fournit en particulier une solution au problème de mots de P et rapproche le monoïde hybride du monoïde de Thompson. Puis, suivant le modèle du monoïde de tresses, on utilise la méthode du retournement de facteur pour analyser la relation de divisibilité à gauche, et montrer en particulier que le monoïde hybride admet la simplification et des ppcm à droite conditionnels. Ensuite, on étudie la combinatoire de Garside de l'hybride: pour chaque entier n, on introduit un élément ∆(n) comme ppcm à droite des (n−1) premiers atomes, et on étudie les diviseurs à gauche des éléments ∆(n), appelés éléments simples. Les principaux résultats sont les dénombrement des diviseurs à gauche de ∆(n) et la détermination effective des formes normales des éléments simples. On termine en construisant des représentations du monoïde hybride dans divers monoïdes, en particulier une représentation dans des matrices à coefficients polynômes de Laurent dont on conjecture qu’elle est fidèle. / We study a certain monoid specified by a presentation, denoted P, that is a hybrid of the classical presentation of the infinite braid monoid and of the presentation of Thompson’s monoid. To this end, we use several approaches. First, we describe a convergent rewrite system for P, which provides in particular a solution to the word problem, and makes the hybrid monoid reminiscent of Thompson’s monoid. Next, on the shape of the braid monoid, we use the factor reversing method to analyze the divisibility relation, and show in particular that the hybrid monoid admits cancellation and conditional right lcms. Then, we study Garside combinatorics of the hybrid: for every integer n, we introduce an element ∆(n) as the right lcm of the first (n−1) atoms, and one investigates the left divisors of the elements ∆(n), called simple elements. The main results are a counting of the left divisors of ∆(n) and a characterization of the normal forms of simple elements. We conclude with the construction of several representations of the hybrid monoid in various monoids, in particular a representation in a monoid of matrices whose entries are Laurent polynomials, which we conjecture could be faithful.

The aim of this thesis is to introduce the reader to the theory of affine monoids and, thereby, to present some results. We therefore start with some auxiliary sections, containing general introductions to convex geometry, affine monoids and their algebras, Hilbert functions and Hilbert series. One central part of the thesis then is the description of an algorithm for computing the integral closure of an affine monoid. The algorithm has been implemented, in the computer program `normaliz´; it outputs the Hilbert basis and the Hilbert function of the integral closure (if the monoid is positive). Possible applications include: finding the lattice points in a lattice polytope, computing the integral closure of a monomial ideal and solving Diophantine systems of linear inequalities. The other main part takes up the notion of multigraded Hilbert function: we investigate the effect of the growth of the Hilbert function along arithmetic progressions (within the grading set) on global growth. This study is motivated by the case of a finitely generated module over a homogeneous ring: there, the Hilbert function grows with a degree which is well determined by the degree of the Hilbert polynomial (and the Krull dimension).

Let Knots be the abelian monoid of isotopy classes of knots S1 ⊂ S3 under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-Orr-Teichner [COT03] defined a filtration of C: C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . .The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5) is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance class of a knot to a pointed set (COT (C/1.5),U). We define an abelian monoid of chain complexes P, with a monoid homomorphism Knots → P. We then define an algebraic concordance equivalence relation on P and therefore a group AC2 := P/ ~, our second order algebraic knot concordance group. The results of this thesis can be summarised in the following diagram: . That is, we define a group homomorphism C → AC2 which factors through C/F(1.5). We can extract the two stage Cochran-Orr-Teichner obstruction theory from AC2: the dotted arrows are morphisms of pointed sets. Our second order algebraic knot concordance group AC2 is a single stage obstruction group.

This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.