Details:
Title  Tree polynomials and nonassociative Gröbner bases  Author(s)  Lothar Gerritzen  Type  Article in Journal  Abstract  In this article the basic notions of a theory of Gröbner bases for ideals in the nonassociative, noncommutative algebra KX 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 KX are called tree polynomials. We describe a criterion for a system of polynomials to constitute a Gröbner basis. It can be seen as a nonassociative version of the Buchberger criterion. A formula is obtained for the generating series of a reduced Gröbner basis for the ideal of nonassociative and noncommutative 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 nonassociative power series like exp , log and the Hausdorff series log(e^xe^y ) and on problems related to Hopf algebras of trees. Reduced Gröbner bases for closed ideals in tree power series algebras KX are considered.  Keywords  Free magmas, Planar binary rooted trees, Nonassociative free algebras, Ideals, Gröbner bases, Nonassociative Buchberger criterion, Composition lemma, Diamond lemma, Hilbert series, Tree power series, Nonassociative exponential and logarithm, Cayley num  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717105001343 
Language  English  Journal  Journal of Symbolic Computation  Volume  41  Number  3–4  Pages  297  316  Year  2006  Note  Logic, Mathematics and Computer Science: Interactions in honor of Bruno Buchberger (60th birthday)  Edition  0  Translation 
No  Refereed 
No 
