A decomposition of elements of the free algebra

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Let f f be an element of F [ X ] F[X] , the free associative algebra over a field F F and n n the maximum of the degrees of the variables and the multiplicities of the degrees in f f . A partial ordering on the homogeneous elements of F [ X ] F[X] is defined such that if f f is homogeneous and F ∤ n ! F\nmid n! , then f f can be decomposed into a sum of two polynomials f 0 {f_0} and f 1 {f_1} such that for 0 > m ⩽ n , f 0 0 > m \leqslant n,\;{f_0} is symmetric or skew symmetric in all its arguments of degree m m depending on whether m m is even or odd and f 1 {f_1} is a consequence of polynomials of lower type than f f . Osborn’s Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The same holds in the free nonassociative algebra. The proofs are combinatorial.