Rational Growth of a Class of Inverse Semigroups

We give a sufficient condition for a finitely presented Rees quotient of a free inverse semigroup to have rational growth. Using related techniques we give a new proof that nonmonogenic free inverse semigroups have irrational growth. A new criterion for polynomial growth is proved and is used to show that polynomial growth implies rational growth. However we give an example of such a semigroup which has rational and exponential growth.