Details:
Title  Nonassociative Gröbner bases  Author(s)  Saeed Rajaee  Type  Article in Journal  Abstract  The theory of Gröbner basis for ideals can be applied in the nonassociative, 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 (α) ∈ 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 nonassociative 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_Γ , the generating series of the reduced Gröbner basis Γ 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 Γ 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 ,ℓ , with fix order i<j<ℓ . Then we obtain the reduced Gröbner basis Γ 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_Γ(t).  Keywords  Free magmas, Planar binary rooted trees, Admissible orders, Multigraded algebras, Nonassociative free algebras, Ideals, Gröbner bases, Rewriting, Hilbert series, Free alternative algebra, Cayley algebra  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S074771710600023X 
Language  English  Journal  Journal of Symbolic Computation  Volume  41  Number  8  Pages  887  904  Year  2006  Edition  0  Translation 
No  Refereed 
No 
