finite semigroup on one generator,cycle, tail,group,zero element

Question

Suppose we have a finite semigroup on one generator. It has a tail of length r and cycle of length c.The cycle is a group, but what can be chosen as a neutral element of it?Why is not ANY element neutral, since we can rotate the cycle?

Answer 1

If I understand your description correctly, the generator $a$ has the property that $a^{r+c}=a^r$. Then, for all elements outside the tail, i.e., for all elements $a^n$ with $n>r$, we can multiply both sides of the equation $a^{r+c}=a^r$ by $a^{n-r}$ to get that $a^na^c=a^{n+c}=a^n$. By induction on $k$, it follows that, for all $n>r$, $a^na^{kc}=a^n$. So the neutral element of the group is $a^{kc}$ for all $k$ that are large enough to put this element into the group (as opposed to being in the tail). (For such $k$, $a^{kc}$ is independent of $k$ because of the $c$ periodicity.)

To answer your concern about "since we can rotate the cycle": Rotating the cycle is likely to tear the tail loose. It's not an automorphism of your semigroup.