# bicyclic semigroup

The bicyclic semigroup $\mathcal{C}({p},{q})$ is the monoid generated by $\{p,q\}$ with the single relation $pq=1$.

The elements of $\mathcal{C}({p},{q})$ are all words of the form $q^{n}p^{m}$ for $m,n\geq 0$ (with the understanding $p^{0}=q^{0}=1$). These words are multiplied as follows:

 $q^{n}p^{m}q^{k}p^{l}=\begin{cases}q^{n+k-m}p^{l}&\text{if }m\leq k,\\ q^{n}p^{l+m-k}&\text{if }m\geq k.\end{cases}$

It is apparent that $\mathcal{C}({p},{q})$ is simple, for if $q^{n}p^{m}$ is an element of $\mathcal{C}({p},{q})$, then $1=p^{n}(q^{n}p^{m})q^{m}$ and so $S^{1}q^{n}p^{m}S^{1}=S$.

It is also easy to see that $\mathcal{C}({p},{q})$ is an inverse semigroup: the element $q^{n}p^{m}$ has inverse $q^{m}p^{n}$.

It is useful to picture some further properties of $\mathcal{C}({p},{q})$ by arranging the elements in a table:

 $\begin{matrix}1&p&p^{2}&p^{3}&p^{4}&\dots\\ q&qp&qp^{2}&qp^{3}&qp^{4}&\dots\\ q^{2}&q^{2}p&q^{2}p^{2}&q^{2}p^{3}&q^{2}p^{4}&\dots\\ q^{3}&q^{3}p&q^{3}p^{2}&q^{3}p^{3}&q^{3}p^{4}&\dots\\ q^{4}&q^{4}p&q^{4}p^{2}&q^{4}p^{3}&q^{4}p^{4}&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}$

Then the elements below any horizontal line drawn through this table form a right ideal and the elements to the right of any vertical line form a left ideal. Further, the elements on the diagonal are all idempotents and their standard ordering is

 $1>qp>q^{2}p^{2}>q^{3}p^{3}>\cdots.$
Title bicyclic semigroup BicyclicSemigroup 2013-03-22 13:09:57 2013-03-22 13:09:57 mclase (549) mclase (549) 8 mclase (549) Definition msc 20M99 bicyclic monoid bicyclic