Canonical Bases for Algebraic Computations

Abstract: This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in non-commutative 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 non-commutative polynomial ring. The second part is a collection of six papers. In the first paper we investigate, for quotients of the non-commutative 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 one-sided ideals, and show a few applications, e.g. how to compute (one-sided) 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 non-commutative 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 bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton 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 bi-automaton 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.