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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.

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dimid
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  • 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 08 '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 08 '16 at 12:20
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    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 08 '16 at 12:22
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    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 08 '16 at 12:22
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    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. – M.B. Jul 08 '16 at 12:25
  • @alex-meiburg What do you mean by "monoids don't have much structure to talk about themselves"? – J.-E. Pin Jul 08 '16 at 12:59

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Your line of thinking is correct: all that is missing from your assumptions to conclude if you have a monoid is to check whether or not the binary operation is associative. This is not true in general, as commutativity doesn't imply associativity, demonstrated many times over in the second of your links.

M.B.
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