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TitleNon-associative Gröbner bases
Author(s) Saeed Rajaee
TypeArticle in Journal
AbstractThe theory of Gröbner basis for ideals can be applied in the non-associative, noncommutative free magma algebra KX with unit freely generated by a set X over a field K . In this article we introduce a class of admissible orders on the magma freely generated by X which is denoted by Mag(X) and some special admissible orders on Mag(X). We prove that the reduced Gröbner basis of a multigraded ideal I in the multigraded algebra KX consists of the reduced multihomogeneous polynomials with multidegrees (&#945;) &#8712; N^n . We obtain a generalization for the Hilbert series of the multigraded algebra A = KX/J of residue classes modulo multigraded ideal J generated by multihomogeneous polynomials in the non-associative free magma algebra of tree polynomials KX , where X is a multigraded set of variables. It relates H A to G X , the generating series in n variables for X , and G_&#915; , the generating series of the reduced Gröbner basis &#915; of J . Let I alt ( X ) be the alternator ideal generated by alternators in the free magma algebra Kx, y, z, then we obtain the elements of multidegree (2, 1, 1) in the reduced Gröbner basis &#915; of I alt ( X ) w.r.t. the admissible order degree first factor on Mag (X). We consider the Cayley algebra O and the admissible order degree first factor on Mag(X), where X = i ,j ,&#8467; , with fix order i<j<&#8467; . Then we obtain the reduced Gröbner basis &#915; of the ideal J generated by all relations in the Cayley algebra O w.r.t. this admissible order. Also we obtain the generating series of G_O(t) and G_&#915;(t).
KeywordsFree magmas, Planar binary rooted trees, Admissible orders, Multigraded algebras, Non-associative free algebras, Ideals, Gröbner bases, Rewriting, Hilbert series, Free alternative algebra, Cayley algebra
URL http://www.sciencedirect.com/science/article/pii/S074771710600023X
JournalJournal of Symbolic Computation
Pages887 - 904
Translation No
Refereed No