I have to figure out the duals to a couple of graded algebras. This requires a comultiplication (also called a coproduct in Hatcher). Hatcher's book shows what form the comultiplication must take using a cohomology argument for the cohomology of H-spaces. I do not know that these algebras are the cohomology of some H-spaces. Is there a purely algebraic argument to figure out a comultipication for an arbitrary graded algebra or do I need some extra data.
If you were wondering, I need to figure the duals of $\Lambda (y)$ and $\Gamma [ \gamma ] $.