Assume there is an element $a\in M$, where $M$ is a non-commutative monoid. If there exists $a b\in M$ such that $a * b = n$, where $n$ is the neutral, does it follow that $b * a = n$?
I want to say no, because the axioms which define a group shouldn't be redundant ($a * b = b * a = n$ is the 4th axiom). But the following logic seems to prove that the inverse is automatically associative:
$b * n = b$
$b * (a * b) = b$
$(b*a) * b = b$
Thus for the last line to be true, $b*a$ has to evaluate to $n$.