Definition 4 (action of [??]Sym on MSym) For w [member of] [[??].sub..] and s [member of] [M.sub.p], write b = [beta](w) and set
This action may be combined with any section of [beta] to define a product on MSym. For example,
Theorem 5 The action [??]Sym [cross product] MSym [right arrow] MSym and the product MSym [cross product] MSym [cross product] MSym are associative.
Unfortunately, no natural coalgebra structure exists on MSym that makes [beta] into a Hopf algebra map.
Definition 6 (action and coaction of ySym on MSym) Given b [member of] M., let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote a p-splitting satisfying [absolute value of [b.sub.0] > 0.
Example 7 In the fundamental bases of MSym and ySym, the action looks like
Theorem 8 The maps * : MSym [cross product] YSym [right arrow] MSym and [rho] : MSym [right arrow] MSym [cross product] YSym are associative and coassociative, respectively.
We next introduce "monomial bases" for [??]Sym, MSym, and YSym.
The monomial basis of MSym demonstrates this isomorphism explicitly.