cyclic semigroup

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

Let $S=⟨x⟩$ 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

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