Canonical Bases for Algebraic Computations
(2001) In Doctoral Theses in Mathematical Sciences Abstract
 This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in noncommutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the noncommutative polynomial ring. The second part is a collection of six papers.
In the first paper we investigate, for quotients of the noncommutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for onesided ideals, and show a few applications, e.g. how to... (More)  This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in noncommutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the noncommutative polynomial ring. The second part is a collection of six papers.
In the first paper we investigate, for quotients of the noncommutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for onesided ideals, and show a few applications, e.g. how to compute (onesided) syzygy modules. The material of the third paper is in some sense related to the contents of this first paper; in the third paper, the theory of SAGBI bases is extended to factor algebras.
The second and fourth paper concerns composition of polynomials. In the first of those two papers, we give sufficient and necessary conditions on a set of polynomials to guarantee that the property of being a noncommutative Gröbner basis is preserved after composition by this set. The latter paper treats the same problem for SAGBI bases.
In the fifth paper we introduce the concept of biautomaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A biautomaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of biautomaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.
In the last paper we investigate various important properties of regular languages associated with quotients of the free associative algebra. We suggest a generalization of a graph for normal words introduced by Ufnarovski, applicable to testing Noetherian properties of automaton algebras. Finally we show an alternative way to compute the generators for the Jacobson radical of any automaton monomial algebra. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/41926
 author
 Nordbeck, Patrik ^{LU}
 supervisor
 opponent

 Prof Apel, Joachim, University of Leipzig, Germany
 organization
 publishing date
 2001
 type
 Thesis
 publication status
 published
 subject
 keywords
 gruppteori, algebra, algebraisk geometri, fältteori, Talteori, group theory, algebraic geometry, field theory, Matematik, Number Theory, regular languages, Mathematics, composition of polynomials, factor algebras, Gröbner bases, SAGBI bases
 in
 Doctoral Theses in Mathematical Sciences
 pages
 194 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 Matematikcentrum, sal C
 defense date
 20011005 10:15:00
 external identifiers

 other:LUTFMA10122001
 ISSN
 14040034
 ISBN
 9162849697
 language
 English
 LU publication?
 yes
 id
 e8baa4f0981d4415bf7c7f8ba50bdb43 (old id 41926)
 date added to LUP
 20160401 17:00:03
 date last changed
 20190521 13:43:15
@phdthesis{e8baa4f0981d4415bf7c7f8ba50bdb43, abstract = {{This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in noncommutative algebras. The material has a natural division into two parts. The first part is a rather extensive treatment of the basic theory of Gröbner bases and SAGBI bases in the noncommutative polynomial ring. The second part is a collection of six papers.<br/><br> <br/><br> In the first paper we investigate, for quotients of the noncommutative polynomial ring, a property that implies finiteness of Gröbner bases computation, and examine its connection with Noetherianity. We propose a Gröbner bases theory for factor algebras, of particular interest for onesided ideals, and show a few applications, e.g. how to compute (onesided) syzygy modules. The material of the third paper is in some sense related to the contents of this first paper; in the third paper, the theory of SAGBI bases is extended to factor algebras.<br/><br> <br/><br> The second and fourth paper concerns composition of polynomials. In the first of those two papers, we give sufficient and necessary conditions on a set of polynomials to guarantee that the property of being a noncommutative Gröbner basis is preserved after composition by this set. The latter paper treats the same problem for SAGBI bases.<br/><br> <br/><br> In the fifth paper we introduce the concept of biautomaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A biautomaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a Gröbner basis regular. We give several examples of biautomaton algebras, and show how automata connected to regular Gröbner bases can be used to perform reduction.<br/><br> <br/><br> In the last paper we investigate various important properties of regular languages associated with quotients of the free associative algebra. We suggest a generalization of a graph for normal words introduced by Ufnarovski, applicable to testing Noetherian properties of automaton algebras. Finally we show an alternative way to compute the generators for the Jacobson radical of any automaton monomial algebra.}}, author = {{Nordbeck, Patrik}}, isbn = {{9162849697}}, issn = {{14040034}}, keywords = {{gruppteori; algebra; algebraisk geometri; fältteori; Talteori; group theory; algebraic geometry; field theory; Matematik; Number Theory; regular languages; Mathematics; composition of polynomials; factor algebras; Gröbner bases; SAGBI bases}}, language = {{eng}}, publisher = {{Centre for Mathematical Sciences, Lund University}}, school = {{Lund University}}, series = {{Doctoral Theses in Mathematical Sciences}}, title = {{Canonical Bases for Algebraic Computations}}, year = {{2001}}, }