Section 6.3
Invertibility in monoids

Definition

In a monoid M with unit element e an element h is called the inverse of g if

g * h = h * g = e.

An element is called invertible if it has an inverse.

We will explain later that the use of the adjective `the' is justified.

If an element is invertible, then we can `divide':

Cancellation law

Let x, y, z be elements of a monoid.

If x is invertible and x * y = x * z, then y = z.