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For every symmetric monoidal category $C$ the category of commutative monoids $\mathrm{CMon}(C)$ is again symmetric monoidal, the forgetful functor to $C$ being a monoidal functor. Thus we may iterate this construction and define $\mathrm{CMon}^{(n)}(C)$ for all $n \in \mathbb{N}$. Intuitively, objects of this category have $n$ commutative operations which are compatible with each other. However, this sequences already terminates at $n \geq 1$. Every commutative monoid in $C$ has a unique structure as an object in $\mathrm{CMon}(C)$.

For example, for $C=\mathrm{Ab}$, starting with abelian groups ($n=0$), next we get commutative rings ($n=1$). But for a commutative ring $R$ there is exactly one unit $u : \mathbb{Z} \to R$ and exactly one ring homomrphism $\mu : R \otimes R \to R$ making it into a commutative monoid of commutative rings, namely $u(z)=z \cdot 1_R$ and $\mu(a \otimes b)=a \cdot b$.

This explains a little bit why there are so few structures with three or more binary operations.

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For every symmetric monoidal category $C$ the category of commutative monoids $\mathrm{CMon}(C)$ is again symmetric monoidal, the forgetful functor to $C$ being a monoidal functor. Thus we may iterate this construction and define $\mathrm{CMon}^{(n)}(C)$ for all $n \in \mathbb{N}$. Intuitively, objects of this category have $n$ commutative operations which are compatible with each other. However, this sequences already terminates at $n \geq 1$. Every commutative monoid in $C$ has a unique structure as an object in $\mathrm{CMon}(C)$.

For example, for $C=\mathrm{Ab}$, starting with abelian groups ($n=0$), next we get commutative rings ($n=1$). But for a commutative ring $R$ there is exactly one unit $u : \mathbb{Z} \to R$ and exactly one ring homomrphism $\mu : R \otimes R \to R$ making it into a commutative monoid of commutative rings, namely $u(z)=z \cdot 1_R$ and $\mu(a \otimes b)=a \cdot b$.