Suppose we have a set $S$ with a single commutative binary operation $*$ and an identity element $i$. Is $(S, *, i)$ necessarily a monoid?
According to this answer, a monoid just requires an associative operation and an identity element, so if I'm not mistaken, the question can be reduced to whether $*$ is necessarily associative. According to this answer, the answer is no.
I'm new to category theory, so I wanted to know if my reasoning is correct.