Invertible implies cancellative in monoid

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Statement

In a monoid (a set with an associative binary operation possessing a multiplicative identity element) the following are true:

Proof

We'll give here the proof for left invertible and left cancellative. An analogous proof works for right invertible and right cancellative.

Given: A monoid M with binary operation *, and identity element (also called neutral element) e. a \in M has a left inverse b (i.e. an element b * a = e)

To prove: a is left-cancellative: whenever c,d \in M are such that a * c = a * d, then c =d

Proof: We start with:

a * c = a * d

Left-multiply both sides by b:

b * (a * c) = b * (a * d)

Use associativity:

(b * a) * c = (b * a) * d

We now use that b * a = e is the identity element, to conclude that c = d.