# adjoining an identity to a semigroup

It is possible to formally adjoin an identity element^{} to any semigroup to make it into a monoid.

Suppose $S$ is a semigroup without an identity^{}, and consider the set $S\cup \{1\}$ where $1$ is a symbol not in $S$. Extend the semigroup operation^{} from $S$ to $S\cup \{1\}$ by additionally defining:

$$s\cdot 1=s=1\cdot s,\text{for all}s\in S\cup 1$$ |

It is easy to verify that this defines a semigroup (associativity is the only thing that needs to be checked).

As a matter of notation, it is customary to write ${S}^{1}$ for the semigroup $S$ with an identity adjoined in this manner, if $S$ does not already have one, and to agree that ${S}^{1}=S$, if $S$ does already have an identity.

Despite the simplicity of this construction, however, it rarely allows one to simplify a problem by considering monoids instead of semigroups. As soon as one starts to look at the structure^{} of the semigroup, it is almost invariably the case that one needs to consider subsemigroups and ideals of the semigroup which do not contain the identity.

Title | adjoining an identity to a semigroup |
---|---|

Canonical name | AdjoiningAnIdentityToASemigroup |

Date of creation | 2013-03-22 13:01:19 |

Last modified on | 2013-03-22 13:01:19 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 5 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M99 |

Related topic | Semigroup |

Related topic | Monoid |