**Properties**

The bicyclic semigroup has the property that the image of any morphism φ from *B* to another semigroup *S* is either cyclic, or it is an isomorphic copy of *B*. The elements φ(*a*), φ(*b*) and φ(*e*) of *S* will always satisfy the conditions above (because φ is a morphism) with the possible exception that φ(*b*) φ(*a*) might turn out to be φ(*e*). If this is not true, then φ(*B*) is isomorphic to *B*; otherwise, it is the cyclic semigroup generated by φ(*a*). In practice, this means that the bicyclic semigroup can be found in many different contexts.

The idempotents of *B* are all pairs (*x*, *x*), where *x* is any natural number (using the ordered pair characterisation of *B*). Since these commute, and *B* is *regular* (for every *x* there is a *y* such that *x* *y* *x* = *x*), the bicyclic semigroup is an inverse semigroup. (This means that each element *x* of *B* has a unique inverse *y*, in the "weak" semigroup sense that *x* *y* *x* = *x* and *y* *x* *y* = *y*.)

Every ideal of *B* is principal: the left and right principal ideals of (*m*, *n*) are

- (
*m*,*n*)*B*= {(*s*,*t*) :*s*≥*m*} and *B*(*m*,*n*) = {(*s*,*t*) :*t*≥*n*}.

Each of these contains infinitely many others, so *B* does not have minimal left or right ideals.

In terms of Green's relations, *B* has only one *D*-class (it is *bisimple*), and hence has only one *J*-class (it is *simple*). The *L* and *R* relations are given by

- (
*a*,*b*)*R*(*c*,*d*) if and only if*a*=*c*; and - (
*a*,*b*)*L*(*c*,*d*) if and only if*b*=*d*.

This implies that two elements are *H*-related if and only if they are identical. Consequently, the only subgroups of *B* are infinitely many copies of the trivial group, each corresponding to one of the idempotents.

The egg-box diagram for *B* is infinitely large; the upper left corner begins:

(0, 0) | (1, 0) | (2, 0) | ... |

(0, 1) | (1, 1) | (2, 1) | ... |

(0, 2) | (1, 2) | (2, 2) | ... |

... | ... | ... | ... |

Each entry represents a singleton *H*-class; the rows are the *R*-classes and the columns are *L*-classes. The idempotents of *B* appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each *L*-class and each *R*-class must contain exactly one idempotent.

The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. Where the definition of *B* from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative lattice under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.

Read more about this topic: Bicyclic Semigroup

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“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)

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