# cyclic semigroup

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

Let $S=\u27e8x\u27e9$ be a cyclic semigroup. Then as a set, $S=\{{x}^{n}\mid n>0\}$.

If all powers of $x$ are distinct, then $S=\{x,{x}^{2},{x}^{3},\mathrm{\dots}\}$ is (countably) infinite.

Otherwise, there is a least integer $n>0$ such that ${x}^{n}={x}^{m}$ for some $$. It is clear then that the elements $x,{x}^{2},\mathrm{\dots},{x}^{n-1}$ 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 $K=\{{x}^{m},{x}^{m+1},\mathrm{\dots},{x}^{n-1}\}$ 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 $X=\{1,2,3,\mathrm{\dots},m+r\}$. Let

$$\varphi =\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill \mathrm{\dots}\hfill & \hfill m+r-1\hfill & \hfill m+r\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill \mathrm{\dots}\hfill & \hfill m+r\hfill & \hfill r+1\hfill \end{array}\right).$$ |

Then $\varphi $ 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 |
---|---|

Canonical name | CyclicSemigroup |

Date of creation | 2013-03-22 13:07:30 |

Last modified on | 2013-03-22 13:07:30 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M99 |

Defines | index |

Defines | period |