We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E*(–). Our description is as a graded and completed version of a Tall-Wraith monoid. The E*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.
Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.75-132.