|
| |
|
| |
|
| |
|
|
Problém realizace von Neumannovsky regulárních okruhů
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
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...
|
|
|
Gröbner bases
Petržilková, Lenka ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we remind you of the basic Buchberger algorithm for com- puting the Gröbner base over commutative polynomial rings. We also observe uniqueness of the Gröbner base for the ideal. Next we research less known, but more effective (for some instances) Faugère F4 algorithm. At the end of the first chapter we compare these two algorithms. In the second chapter we analyze a generalization of the Buchberger algorithm for noncommutative rings both for free algebra and factor algebra. On the contary to the commu- tative case, Gröbner bases can be infinite in this case, even for some finitely generated ideals. Among other things, we investigate quasi-zero elements,i.e. such elements, that we get zero by multiplying them with an arbitrary term, and their role in the division of a polynom by set of polynoms. 1
|
|
| |
|
| |
|
| |
|
|
Algebraický přístup k CSP
Bulín, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
For a finite relational structure A, the Constraint Satisfaction Problem with template A, or CSP(A), is the problem of deciding whether an input relational structure X admits a homomorphism to A. The CSP dichotomy conjecture of Feder and Vardi states that for any A, CSP(A) is either in P or NP-complete. In the first part we present the algebraic approach to CSP and summarize known results about CSP for digraphs, also known as the H-coloring problem. In the second part we study a class of oriented trees called special polyads. Using the algebraic approach we confirm the dichotomy conjecture for special polyads. We provide a finer description of the tractable cases and give a construction of a special polyad T such that CSP(T) is tractable, but T does not have width 1 and admits no near-unanimity polymorphisms.
|
|
|
An algorithmic approach to resolutions in representation theory
Ivánek, Adam ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the- ory of finite-dimensional algebras. In this thesis finite-dimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQ-modules and the theory described in article Minimal Pro- jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri- vial K-linear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1
|
guest ::
RSS feed.