In this paper we generalize some basic applications of Grr obner bases in commutative polynomial rings to the non-commutative case. We deene a non-commutative elimination order. Methods of nding the intersection of two ideals are given. If both the ideals are monomial we deduce a nitely written basis for their intersection. We nd the kernel of a… (More)
Canonical bases, also called SAGBI bases, for subalgebras of the non-commutative polynomial ring are investigated. The process of subalgebra reduction is deened. Methods, including generalizations of the standard Grr obner bases techniques , are developed for the test whether bases are canonical, and for the completion procedure of constructing canonical… (More)
Polynomial composition is the operation of replacing the variables in a polynomial with other polynomials. In this paper we give a sufficient and necessary condition on a set Θ of polynomials to assure that the set F • Θ of composed polynomials is a SAGBI basis whenever F is.
Polynomial composition is the operation of replacing the variables in a polynomial with other polynomials. In this paper we give sufficient and necessary conditions on a set Y of non-commutative polynomials to assure that the set G + Y of composed polynomials is a Gro¨bner basis in the free asso-ciative algebra whenever G is. The subject was initiated by… (More)
We introduce canonical bases for subalgebras of quotients of the commutative and non-commutative polynomial ring. The usual theory for Gröbner bases and its counterpart for subalge-bras of polynomial rings, also called SAGBI bases, are combined to obtain a tool for computation in subalgebras of factor algebras.
Preface " Trying is the first step towards failure. " Homer Simpson This thesis deals with computational methods in algebra, mainly focusing on the concept of Gröbner and SAGBI bases in non-commutative algebras. The theory of these bases is constructive in the meaning that the purpose is to provide methods for solving specific problems. As the reader will… (More)
We introduce canonical bases for subalgebras of quotients of the commutative and non-commutative polynomial ring. A more complete exposition can be found in 4]. Canonical bases for subalgebras of the commutative polynomial ring were introduced by Kapur and Madlener (see 2]), and independently by Robbiano and Sweedler ((5]). Some notes on the non-commutative… (More)
In this 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… (More)