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 $𝔉(X)$ the set of partial map on $X$. Given $\alpha \in 𝔉(X)$, 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 $\alpha ,\beta \in 𝔉(X)$ as the partial map $\alpha \circ \beta \in 𝔉(X)$ 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 $𝔉(X)$ with the composition $\circ$ is a semigroup.

A partial map $\alpha \in 𝔉(X)$ 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 $\Im (X)\subseteq 𝔉(X)$ 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 $\Im (X)$.