# Sufficient condition for a monoid

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.

• I don't see what this has to do with category theory. And I also don't see how your question is not answered by the linked answer. – Tobias Kildetoft Jul 8 '16 at 12:16
• @TobiasKildetoft isn't monoid a part of category theory? If not, what field of math does it belong to? I guess it's answered, I just wanted to verify my reasoning. – dimid Jul 8 '16 at 12:20
• Monoids belong to abstract algebra. Part of the definition of a category means that the set of arrows from an object to itself forms a monoid, but that does not mean that monoids are a part of category theory. There is also a more complicated notion of a "monoid object" in a category, but that is probably not what you are looking for. – Tobias Kildetoft Jul 8 '16 at 12:22
• A monoid is really considered more an element of abstract algebra (albeit, a less structured part), since it's vaguely "group-like" (an associative operation). That said, monoids don't have much structure to talk about themselves, so they're most interesting to category theorists. Edit: Darn ninjas. ;) – Alex Meiburg Jul 8 '16 at 12:22
• Indeed monoids are a thing in category theory, don't worry. In fact (referring to Awodey's book), a monoid is a category with just one object. The actual elements of the set $S$ are then the arrows of this category, and in particular the identity element becomes the identity arrow. Composition of arrows in this category is then applications of the monoid operator. – MonadBoy Jul 8 '16 at 12:25