# semigroup with two elements

Perhaps the simplest non-trivial example of a semigroup which is not a group is a particular semigroup with two elements. The underlying set of this
semigroup is $\{a,b\}$ and the operation^{} is defined as follows:

$a\cdot a$ | $=$ | $a$ | ||

$a\cdot b$ | $=$ | $b$ | ||

$b\cdot a$ | $=$ | $b$ | ||

$b\cdot b$ | $=$ | $b$ |

It is rather easy to check that this operation is associative, as it should be:

$a\cdot (a\cdot a)=a\cdot a=$ | $a$ | $=a\cdot a=(a\cdot a)\cdot a$ | ||

$a\cdot (a\cdot b)=a\cdot b=$ | $b$ | $=a\cdot b=(a\cdot a)\cdot b$ | ||

$a\cdot (b\cdot b)=a\cdot b=$ | $b$ | $=b\cdot b=(a\cdot b)\cdot b$ | ||

$b\cdot (a\cdot a)=b\cdot a=$ | $b$ | $=a\cdot a=(a\cdot a)\cdot a$ | ||

$a\cdot (b\cdot b)=a\cdot b=$ | $b$ | $=b\cdot b=(a\cdot b)\cdot b$ | ||

$b\cdot (a\cdot b)=b\cdot b=$ | $b$ | $=b\cdot b=(b\cdot a)\cdot b$ | ||

$b\cdot (b\cdot a)=b\cdot b=$ | $b$ | $=b\cdot a=(b\cdot b)\cdot a$ | ||

$b\cdot (b\cdot b)=b\cdot b=$ | $b$ | $=b\cdot b=(b\cdot b)\cdot b$ |

It is worth noting that this semigroup is commutative^{} and has an identity
element^{}, which is $a$. It is not a group because the element $b$ does
not have an inverse^{}. In fact, it is not even a cancellative semigroup
because we cannot cancel the $b$ in the equation $a\cdot b=b\cdot b$.

This semigroup also arises in various contexts. For instance, if we choose $a$ to be the truth value ”true” and $b$ to be the truth value ”false” and the operation $\cdot $ to be the logical connective ”and”, we obtain this semigroup in logic. We may also represent it by matrices like so:

$$a=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\mathit{\hspace{1em}\hspace{1em}}b=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$$ |

Title | semigroup with two elements |
---|---|

Canonical name | SemigroupWithTwoElements |

Date of creation | 2013-03-22 16:21:42 |

Last modified on | 2013-03-22 16:21:42 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Example |

Classification | msc 20M99 |