Group of units of a monoid

Definition

Let $(M, \ast, e)$ be a monoid. An element $u \in M$ is a unit when it has a two-sided inverse, that is, an element $v \in M$ with $u \ast v = v \ast u = e$. The group of units of $M$ is the set $M^\times$ of the units of $M$, with the operation $\ast$ restricted to $M^\times$. It is a group with identity $e$.

Remarks

The units of the monoid $(\mathbb{Z}, \cdot)$ are $1$ and $-1$. The units of the monoid of $n \times n$ real matrices under multiplication are the invertible matrices, and they form the general linear group $\mathrm{GL}_n(\mathbb{R})$. For a ring $R$, the group of units $R^\times$ is the group of units of the monoid $(R, \cdot)$.

Explanations

L1 human

In a monoid, some elements can be undone and others cannot. The elements that can be undone are the units. The units alone form a group, because products and inverses of units are units.