2019/05/31 by Jordi Delgado, Pedro V. Silva
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Algebra over a field #Automaton #Finitely-generated abelian group #Geometric and Algebraic Topology #Intersection (aeronautics) #Lattice (music) #Rank (graph theory) #math.GR
paper · pdf · doi:10.46298/jgcc.2020.12.1.6059
published as journal of Groups, complexity, cryptology, Volume 12, Issue 1 (March 2, 2020) gcc:6059 · 27 pages, 5 figures
openalex created_date 2019/06/07 · arxiv created 2020/02/28 · openalex publication_date 2020/03/02 · arxiv updated 2022/03/14 · openalex updated_date 2026/08/05
A \vee-complement of a subgroup H \leqslant \mathbbFn is a subgroup K \leqslant \mathbbFn such that H \vee K = \mathbbFn. 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 \vee-complement (resp. ⊕-complement) of H is called the \vee-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 \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.