Building factor semigroup w.r.t. $I_n = \{ w \in A^+ : |w| \ge n \}$.

Question

I don't understand the following factor semigroup.

Consider the pseudovariety $N$ of nilpotent semigroups. For any finite alphabet $A$, let $I_n = \{ w \in A^+ : |w| \ge n \}$. Then $A^+ / I_n$ is a nilpotnet semigroup; thus $A^+ / I_n \in N$. Thus $N$ contains arbitrarily large $A$-generated semigroups.

I don't see why $A^+ / I_n$ gets arbitrarily large, because the only factor structur I can image regarding $I_n$ is to divide in those words $w$ with $|w| \ge n$ and those with $|w| < n$, so I always get two classes, i.e. $|A^+ / I_n| = 2$ and not arbitrary large. Can someone explain this to me, maybe the congruence relation derived from $I_n$ to construct the factor structur is another as $u \sim v :\Leftrightarrow u,w \in I_n$?

By the way, the example is taken from these lecture notes:

http://www-groups.mcs.st-andrews.ac.uk/~alanc/teaching/m431/index.html

Answer 1

Correction to my previous answer (I misread $|w| \geq n$).

You should first read the section about Rees factor semigroups (bottom of page 16 and page 17 in the lecture notes you give in reference). Let me just quote a sentence from these notes:

It is often convenient to view $S/I$ as having elements $(S \setminus I) \cup \{0\}$, and thinking of forming $S/I$ by starting with $S$ and merging all elements of $I$ to form a zero.

Thus the elements of $A^+/I_n$ are $(A^{<n})\cup \{0\}$, where I denote by $A^{<n}$ the set of all nonempty words of length $< n$. If $|A| > 1$, the size of $A^+/I_n$ is $$ \frac{|A|^n - 1}{|A|-1} $$ and if $|A| = 1$, its size is $n$. Thus, for any nonempty $A$, the pseudovariety of finite nilpotent semigroups contains arbitrarily large $A$-generated semigroups.

Answer 2

$A^+ / I_n$ gets arbitrarily large if we take $n$ arbitrarily large. Further, let $F$ be an A-generated free object, then $F^m=0$ for some $m$. Since $A^+ / I_n=\langle A\rangle$, $F$ must map surjectively onto $A^+ / I_n$ for every $n$. It is impossible since $|F|<\infty$.