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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 havesuperexponential growth of strict type $n^n$ for finite free rank greater than1. It is also shown that free semigroups have the fastest subsemigroup growthof all finitely generated semigroups. Ideal growth is shown to be exponentialwith strict type $2^n$ and congruence growth is shown to be at leastexponential. In addition we consider the case when the index is fixed and rankincreasing, proving that for subsemigroups and ideals this sequence fits apolynomial of degree the index, whereas for congruences this fits anexponential equation of base the index. We use these results to describe analgorithm for computing values of these sequences and give a table of resultsfor low rank and index.

Subsemigroup, ideal and congruence growth of free semigroups