THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and de. ne the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we de. ne orders on the set partitions and set compositions which allow us to de. ne bases which have simple and natural rules for the product of basis elements.