# On the lattice of subgroups of a free group: complements and rank

### journal of Groups, Complexity, Cryptology

#### Article

A ∨-complement of a subgroup H⩽Fn is a subgroup K⩽Fn such that H∨K=Fn. If we also ask K to have trivial intersection with H, then we say that K is a ⊕-complement of H. The minimum possible rank of a ∨-complement (resp. ⊕-complement) of H is called the ∨-corank (resp. ⊕-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 ∨-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.

### Publication

Year of publication: 2020

Volume: Volume 12

Issue: 1

Date published: 02/2020

Other: gcc:6059