10.46298/jgcc.2020.12.1.6059 Delgado, Jordi Jordi Delgado Silva, Pedro V. Pedro V. Silva On the lattice of subgroups of a free group: complements and rank episciences.org 2020 Mathematics - Group Theory contact@episciences.org episciences.org 2020-01-28T18:15:32+01:00 2020-03-02T12:56:10+01:00 2020-03-02 eng Journal article https://gcc.episciences.org/6059 arXiv:1905.12597 1869-6104 PDF 1 journal of Groups, complexity, cryptology ; Volume 12, Issue 1 ; 1869-6104 A \$\vee\$-complement of a subgroup \$H \leqslant \mathbb{F}_n\$ is a subgroup \$K \leqslant \mathbb{F}_n\$ such that \$H \vee 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 \$\vee\$-complement (resp. \$\oplus\$-complement) of \$H\$ is called the \$\vee\$-corank (resp. \$\oplus\$-corank) of \$H\$. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the \$\vee\$-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds. Comment: 27 pages, 5 figures