Subsemigroup, ideal and congruence growth of free semigroups

TL;DR

This paper investigates the growth rates of subsemigroups, ideals, and congruences in free semigroups, revealing superexponential, exponential, and at least exponential growth patterns, and provides algorithms for computing these sequences.

Contribution

It establishes precise growth bounds for subsemigroups, ideals, and congruences in free semigroups, including superexponential and polynomial growth rates, and introduces algorithms for their computation.

Findings

Subsemigroup growth is superexponential of type n^n for rank > 1.

Ideal growth is exponential with type 2^n.

Congruence growth is at least exponential.

Abstract

Using Rees index, the subsemigroup growth of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type $n^n$ for finite free rank greater than 1. It is also shown that free semigroups have the fastest subsemigroup growth of all finitely generated semigroups. Ideal growth is shown to be exponential with strict type $2^n$ and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequence fits a polynomial of degree the index, whereas for congruences this fits an exponential equation of base the index. We use these results to describe an algorithm for computing values of these sequences and give a table of results for low rank and index.