18,334 research outputs found
On the lattice of subgroups of a free group: complements and rank
A -complement of a subgroup is a subgroup such that . If we also ask
to have trivial intersection with , then we say that is a
-complement of . The minimum possible rank of a -complement
(resp. -complement) of is called the -corank (resp.
-corank) of . 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.Comment: 27 pages, 5 figure
On periodic points of free inverse monoid endomorphisms
It is proved that the periodic point submonoid of a free inverse monoid
endomorphism is always finitely generated. Using Chomsky's hierarchy of
languages, we prove that the fixed point submonoid of an endomorphism of a free
inverse monoid can be represented by a context-sensitive language but, in
general, it cannot be represented by a context-free language.Comment: 18 page
- …