p-perfection and group completion of E∞-monoids

We study E∞ -monoids on which a prime p acts invertibly, which we call p -perfect, in the non-group-complete situation. In particular, we prove that in many examples, they almost embed in their group-completion.We further study the p -perfection functor, and describe it in terms of Quillen’s + -construction, similarly to group-completion. This gives an alternative description of the p -inverted higher algebraic K -theory of a ring.