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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

Identifiers

Other: gcc:6059

Alternative Titles

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