The growth of free inverse monoids

TL;DR

This paper calculates the exponential growth rate of free inverse monoids of rank r, providing explicit formulas and bounds, and analyzes the growth of idempotents, revealing asymptotic behaviors.

Contribution

It introduces the exact algebraic number for the growth rate of free inverse monoids and the explicit growth rate of idempotents, advancing understanding of their algebraic structure.

Findings

Growth rate is an algebraic number between 2r-1 and 2r

Explicit expression for the growth rate of idempotents

Asymptotic behavior of growth rates as rank increases

Abstract

We compute the rate of exponential growth of the free inverse monoid of rank $r$ (and hence an upper bound on the corresponding rate for all $r$-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt{e(2k-1)}$ as $k \to \infty$.