Is "(a * a') is cancellative" + "M has an identity" the same as "a has an inverse"

Question

Given a magma $(M, \ast)$, $(a \ast a')$ is cancellative, iff

$$\forall b,c \in M. b \ast (a \ast a') = c \ast (a \ast a')\Leftrightarrow b = c$$

The magma has an identity, iff: $$\exists e.\forall a \in M. a \ast e = a = a \ast e$$

Now if we use the first formula, and use $e$ for b and c, we get: $$e \ast (a \ast a') = e \ast (a \ast a')\Leftrightarrow e = (a \ast a')$$

Is this correct?

Answer 1

No: you have substituted $a*a'$ instead of $e$ for $c$ in the final part. The correct statement is $$e \ast (a \ast a') = e \ast (a \ast a')\Leftrightarrow e = e.$$ This is true, but not interesting (both sides of the $\Leftrightarrow$ are obviously true).

Indeed, your statement is typically false: for instance, the magma $(\mathbb{N},+)$ has an identity and every element is cancellative, but not every element is equal to the identity.