# cyclic semigroup

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 $S=\{x,x^{2},x^{3},\ldots\}$ is (countably) infinite.

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

Title cyclic semigroup CyclicSemigroup 2013-03-22 13:07:30 2013-03-22 13:07:30 mclase (549) mclase (549) 6 mclase (549) Definition msc 20M99 index period