$p$-perfection and group completion of ${\mathbb{E}}_{\mathrm{\infty }}$-monoids

We study ${\mathbb{E}}_{\mathrm{\infty }}$-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.