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A non-connected neither of finite type Hopf algebra $\mathcal{F}_{0}$ isdefined over $\mathbb{Z}/ 2\mathbb{Z}$ and its hom dual turns out to be atensor product of polynomial algebras. Certain quotient Hopf algebras includethe Steenrod and Dyer-Lashof algebras. This setting provides a map between theSteenrod coalgebra and a direct limit of Dyer-Lashof coalgebras.

The $mod2$ Steenrod and Dyer-Lashof algebras as quotients of a free algebra