A semigroup which is generated by a single element is called a cyclic semigroup.

Let $S = \langle x \rangle$ be a cyclic semigroup. Then as a set, $S = \{ x^n \mid n > 0\}$ .

If all powers of $x$ are distinct, then is (countably) infinite.

Otherwise, there is a least integer $n > 0$ such that $x^n = x^m$ for some $m < n$ . It is clear then that the elements $x, x^2, \dots, x^{n-1}$ are distinct, but that for any $j \ge n$ , we must have $x^j = x^i$ for some $i$ , $m \le i \le n-1$ . So $S$ has $n-1$ elements.

Unlike in the group case, however, there are in general multiple non-isomorphic cyclic semigroups with the same number of elements. In fact, there are $t$ non-isomorphic cyclic semigroups with $t$ elements: these correspond to the different choices of $m$ in the above (with $n = t+1$ ).

The integer $m$ is called the index of $S$ , and $n-m$ is called the period of $S$ .

The elements $K = \{x^m, x^{m+1}, \dots, x^{n-1}\}$ are a subsemigroup of $S$ . In fact, $K$ is a cyclic group.

A concrete representation of the semigroup with index $m$ and period $r$ as a semigroup of transformations can be obtained as follows. Let $X = \{1, 2, 3, \dots, m + r\}$ . Let $$ \phi = \begin{pmatrix} 1 & 2 & 3 & \dots & m+r-1 & m+r \\ 2 & 3 & 4 & \dots & m+r & r+1 \end{pmatrix}. $$

Then $\phi$ generates a subsemigroup $S$ of the full semigroup of transformations $\mathcal{T}_X$ , and $S$ is cyclic with index $m$ and period $r$ .