Arnon completion of the Dyer-Lashof algebra

In his PhD thesis, Arnon [1] builds a completion of the Dickson algebras wh ich contains a "free root" algebra D-f (in) on the top Dickson classes. Hu' ng [5] has shown that this algebra is in fact isomorphic to a similar compl etion (A(mu))* of the dual of the Steenrod algebra A*. Amen also completed the Steenrod algebra A with respect to its halving homomorphism to obtain A (mu). Here we study an analogous completion of the Dyer-Lashof algebra R to obtain R-mu with canonical subcoalgebras R-mu[n]. Unlike the Steenrod alge bra, we may further complete R-mu with respect to length to obtain (R) over cap(mu). It turns out, somewhat surprisingly, that the dual ((R) over cap( mu))* contains (A(mu))* as a dense subalgebra.