---
tag: T49YXARXF6
kind: concept
name: Symmetric inverse semigroup
aliases: [partial-symmetries, partial-symmetry]
sources:
  - wikipedia: en:18442482@r1219830941
    locator: Symmetric inverse semigroup
    note: adopted
    mention: wikipedia://en/18442482
    sections: [Summary]
same_as:
  wikidata: Q7661309
  wikipedia: Symmetric inverse semigroup
inventory: [K8Y0XJWFKR]
---

## Summary

In abstract algebra, the [[set|0BSJ7RQ9BW]] of all partial bijections on a set *X* ( one-to-one partial transformations) forms an inverse semigroup, called the **symmetric inverse semigroup** (actually a [[monoid|0N6R902FG1]]) on *X*. The conventional notation for the symmetric inverse semigroup on a set *X* is $\mathcal{I}_X$ or $\mathcal{IS}_X$. In general $\mathcal{I}_X$ is not commutative.