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.
Andrew Stacey. Sarah Whitehouse. "The hunting of the Hopf ring." Homology Homotopy Appl. 11 (2) 75 - 132, 2009.