On the lattice of subgroups of a free group: complements and rank
A $ee$-complement of a subgroup $H leqslant mathbb{F}_n$ is a subgroup $K leqslant mathbb{F}_n$ such that $H ee K = mathbb{F}_n$.If we also ask $K$ to have trivial intersection with $H$, then we say that $K$ is a $oplus$-complement of $H$.The minimum possible rank of a $ee$-complement (resp.$oplus$-complement) of $H$ Roll On is called the $ee$-