CyclicSemigroup 2002-11-01 19:29:43 2002-11-02 20:06:30 Definition index period % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A semigroup which is generated by a single element is called a \emph{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 $S = \{x, x^2, x^3, \dotsc \}$ 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 \emph{index} of $S$, and $n-m$ is called the \emph{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 $r$.