Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Computing Gröbner bases in monoid and group rings
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
An introduction to commutative and noncommutative Gro¨bner bases
Selected papers of the second international colloquium on Words, languages and combinatorics
Constructing bases of finitely presented Lie algebras using Gröbner bases in free algebras
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Non-associative gröbner bases, finitely-presented lie rings and the engel condition
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II
Journal of Symbolic Computation
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In this article the basic notions of a theory of Grobner bases for ideals in the non-associative, non-commutative algebra K{X} with a unit freely generated by a set X over a field K are discussed. The monomials in this algebra can be identified with the set of isomorphism classes of X-labelled finite, planar binary rooted trees where X is the set of free algebra generators. The elements of K{X} are called tree polynomials. We describe a criterion for a system of polynomials to constitute a Grobner basis. It can be seen as a non-associative version of the Buchberger criterion. A formula is obtained for the generating series of a reduced Grobner basis for the ideal of non-associative and non-commutative relations of an algebra relative to a system of algebra generators and an admissible order on the monomials. If the algebra is graded it specializes to a general Hilbert series formula in terms of generators and relations. We also report on new results concerning non-associative power series like exp,log and the Hausdorff series log(e^xe^y) and on problems related to Hopf algebras of trees. Reduced Grobner bases for closed ideals in tree power series algebras K{{X}} are considered.