Non-associative Gröbner bases

The theory of Gröbner basis for ideals can be applied in the non-associative, noncommutative free magma algebra K{X} 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 and some special admissible orders on . We prove that the reduced Gröbner basis of a multigraded ideal I in the multigraded algebra K{X} consists of the reduced multihomogeneous polynomials with multidegrees . We obtain a generalization for the Hilbert series of the multigraded algebra A=K{X}/J of residue classes modulo multigraded ideal J generated by multihomogeneous polynomials in the non-associative free magma algebra of tree polynomials K{X}, where X is a multigraded set of variables. It relates HA to GX, the generating series in n variables for X, and GΓ, the generating series of the reduced Gröbner basis ΓofJ. Let be the alternator ideal generated by alternators in the free magma algebra K{x,y,z}, then we obtain the elements of multidegree (2, 1, 1) in the reduced Gröbner basis Γ of w.r.t. the admissible order degree first factor on . We consider the Cayley algebra and the admissible order degree first factor on , where X={i,j,ℓ}, with fix order i