Tall-Wraith monoids

Tall–Wraith monoids were introduced in [SW09] to describe the alge-braic structure on the set of unstable operations of a suitable generalised co-homology theory. In this paper we begin the study of Tall–Wraith monoids in an algebraic and categorical setting. We show that forV a variety of al-gebras, applying the freeV-algebra functor to a monoid in Set produces a Tall–Wraith monoid. We also study the example of the Tall–Wraith monoid defined by the self set-maps of a finite ring, an example closely related to the original motivation for Tall–Wraith monoids.