Rational growth of a class of inverse semigroups
This is the first of two consecutive papers in this journal (for the second paper see the following review Zbl 0910.20044) on growth functions of inverse semigroups. Let \(S\) be a semigroup generated by a finite set \(X\) and let \(\ell(t)\) denote the minimal length of \(t\in S\) over \(X\). The author works with the power series \(h_S(z)\) where the coefficient of \(z^m\) is \(h_S(m)=|\{t\in S:\ell(t)=m\}|\). The semigroup \(S\) is said to have rational growth if \(h_S(z)\) is a rational function of \(z\). We say that \(S\) has polynomial growth [resp. exponential growth] if \(h_S(m)\) is bounded above by \(m^d\) (some \(d\geq 1)\) [resp. bounded below by \(\alpha^m\) (some \(\alpha>1\))]. The author first proves in a new way a known result: the free inverse semigroup on a finite generating set of order at least two has irrational growth. A condition for a Rees quotient \(S\) of a finitely generated free inverse semigroup in terms of restrictions on cycles of a certain graph associated with \(S\) is sufficient to ensure rational growth. It is proved that, within this class of semigroups, polynomial growth implies rational growth, but the converse is false.
finitely generated free inverse semigroups
scientific article; zbMATH DE number 1218667
scientific article