  Takahasi's theorem on chains of subgroups of bounded rank in a free group is
generalized to several classes of semigroups. As an application, it is proved
that the subsemigroups of periodic points are finitely generated and periodic
orbits are bounded for arbitrary endomorphisms for various semigroups. Some of
these results feature classes such as completely simple semigroups, Clifford
semigroups or monoids defined by balanced one-relator presentations. In
addition to the background on semigroups, proofs involve arguments over groups
and finite automata.
