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

The elements of $\Bicyc{p}{q}$ 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:

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

It is also easy to see that $\Bicyc{p}{q}$ is an inverse semigroup: the element $q^np^m$ has inverse $q^mp^n$ .

It is useful to picture some further properties of $\Bicyc{p}{q}$ by arranging the elements in a table:

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 $$1 > qp > q^2 p^2 > q^3 p^3 > \dotsb.$$