# symmetric inverse semigroup

Let $X$ be a set. A *partial map* on $X$ is an application defined from a subset of $X$ into $X$. We denote by $\U0001d509(X)$ the set of partial map on $X$. Given $\alpha \in \U0001d509(X)$, we denote by $\mathrm{dom}(\alpha )$ and $\mathrm{ran}(\alpha )$ respectively the domain and the range of $\alpha $, i.e.

$$\mathrm{dom}(\alpha ),\mathrm{ran}\alpha \subseteq X,\alpha :\mathrm{dom}(\alpha )\to X,\alpha (\mathrm{dom}(\alpha ))=\mathrm{ran}(\alpha ).$$ |

We define the composition^{} of two partial map $\alpha ,\beta \in \U0001d509(X)$ as the partial map $\alpha \circ \beta \in \U0001d509(X)$ with domain

$$\mathrm{dom}(\alpha \circ \beta )={\beta}^{-1}(\mathrm{ran}(\beta )\cap \mathrm{dom}(\alpha ))=\{x\in \mathrm{dom}(\beta )|\alpha (x)\in \mathrm{dom}(\beta )\}$$ |

defined by the common rule

$$\alpha \circ \beta (x)=\alpha (\beta (x)),\forall x\in \mathrm{dom}(\alpha \circ \beta ).$$ |

It is easily verified that the $\U0001d509(X)$ with the composition $\circ $ is a semigroup.

A partial map $\alpha \in \U0001d509(X)$ is said *bijective* when it is bijective as a map $\alpha :\mathrm{ran}(\alpha )\to \mathrm{dom}(\alpha )$. It can be proved that the subset $\Im (X)\subseteq \U0001d509(X)$ of the partial bijective maps on $X$ is an inverse semigroup (with the composition $\circ $), that is called *symmetric inverse semigroup* on $X$. Note that the symmetric group^{} on $X$ is a subgroup^{} of $\Im (X)$.

Title | symmetric inverse semigroup |
---|---|

Canonical name | SymmetricInverseSemigroup |

Date of creation | 2013-03-22 16:11:14 |

Last modified on | 2013-03-22 16:11:14 |

Owner | Mazzu (14365) |

Last modified by | Mazzu (14365) |

Numerical id | 6 |

Author | Mazzu (14365) |

Entry type | Definition |

Classification | msc 20M18 |

Defines | partial map |

Defines | composition of partial maps |

Defines | symmetric inverse semigroup |