Let $S$ be a finite semigroup of order $n$. Suppose that $S$ has index $m$ and period $r$, i.e. $S$ satisfies the identity $x^{m+r} = x^m$. Then it is quite easy to show that $m \leq n$. My question is, how are $r$ and $n$ related? More specifically, is $r$ bounded by some (polynomial) function $f(n)$ that depends on $n$?
Thank you.
There is no polynomial bound in n for the period r. Let $p_i$ be the $i$-th prime number and let $C_{p_i}$ be the cyclic group of order $p_i$. Now take the disjoint union $S_k$ of $C_{p_1}, \ldots, C_{p_k}$ and $\{0\}$. Make it a commutative semigroup by setting $x + y = 0$ whenever $x$ and $y$ are not in the same cyclic group. Then the order of $S_k$ is $p_1 + \dotsm + p_k + 1$, but its period is $r = p_1 \dotsm p_k$.
