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 H_A to G_X, the generating series in n variables for X, and G_Γ, the generating series of the reduced Gröbner basis Γof J.

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<j<ℓ. Then we obtain the reduced Gröbner basis Γ of the ideal J generated by all relations in the Cayley algebra w.r.t. this admissible order. Also we obtain the generating series of and G_Γ(t).