PlanetMath: symmetric inverse semigroup

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symmetric inverse semigroup

(Definition)

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

$\displaystyle \mathrm{dom}(\alpha),\mathrm{ran}{\alpha}\subseteq X,\ \ \alpha:\... ...om}(\alpha)\rightarrow X,\ \ \alpha(\mathrm{dom}(\alpha))=\mathrm{ran}(\alpha).$

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

$\displaystyle \mathrm{dom}(\alpha\circ\beta)=\beta^{-1}(\mathrm{ran}(\beta)\cap... ...left\{ x\in\mathrm{dom}(\beta)\,\vert\,\alpha(x)\in\mathrm{dom}(\beta) \right\}$

defined by the common rule

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

It is easily verified that the $\mathfrak{F}(X)$ with the composition $\circ$ is a semigroup.

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