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bicyclic semigroup (Definition)

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:

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

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^1q^n p^mS^1 = S$.

It is also easy to see that $ \mathcal{C}({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 $ \mathcal{C}({p},{q})$ by arranging the elements in a table:

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

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

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



"bicyclic semigroup" is owned by mclase.
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Other names:  bicyclic monoid
Also defines:  bicyclic
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Cross-references: ordering, idempotents, diagonal, left ideal, right, right ideal, line, properties, inverse, inverse semigroup, easy to see, simple, words, relation, generated by, monoid
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This is version 5 of bicyclic semigroup, born on 2002-11-19, modified 2004-06-17.
Object id is 3609, canonical name is BicyclicSemigroup.
Accessed 3981 times total.

Classification:
AMS MSC: 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
Pending Errata and Addenda
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