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TitleTree polynomials and non-associative Gröbner bases
Author(s) Lothar Gerritzen
TypeArticle in Journal
AbstractIn this article the basic notions of a theory of Gröbner bases for ideals in the non-associative, non-commutative 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 non-associative version of the Buchberger criterion. A formula is obtained for the generating series of a reduced Gröbner 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 Gröbner bases for closed ideals in tree power series algebras KX are considered.
KeywordsFree magmas, Planar binary rooted trees, Non-associative free algebras, Ideals, Gröbner bases, Non-associative Buchberger criterion, Composition lemma, Diamond lemma, Hilbert series, Tree power series, Non-associative exponential and logarithm, Cayley num
ISSN0747-7171
URL http://www.sciencedirect.com/science/article/pii/S0747717105001343
LanguageEnglish
JournalJournal of Symbolic Computation
Volume41
Number34
Pages297 - 316
Year2006
NoteLogic, Mathematics and Computer Science: Interactions in honor of Bruno Buchberger (60th birthday)
Edition0
Translation No
Refereed No
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