bicyclic semigroup

The bicyclic semigroup $𝒞(p,q)$ is themonoid generated by $\{p,q\}$ withthe single relation $pq=1$.

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

It is apparent that $𝒞(p,q)$ is simple, for if$q^n{p}^m$ is an element of $𝒞(p,q)$, 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 $𝒞(p,q)$ 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 $𝒞(p,q)$ byarranging the elements in a table:

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

$$1>qp>q^2{p}^2>q^3{p}^3>\mathrm{\cdots }.$$