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This is probably a stupid question but could there be an algebraic structure like a group or a ring or something else, with more than $2$ internal laws? like $(G,+,\cdot,\star)$?

I know we could create an additional law on $\Bbb Z$ for example by defining $a\star b=a+b-ab$ or something but that uses the two already existing laws...

Do we study $(\Bbb F[x],+,\cdot,\circ)$ as a structure addition multiplication and composition of polynomials?

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  • $\begingroup$ An algebra has three operations. $\endgroup$ May 23, 2018 at 9:08
  • $\begingroup$ Vector spaces ? $\endgroup$
    – user65203
    May 23, 2018 at 10:01
  • $\begingroup$ @YvesDaoust How? $\endgroup$ May 23, 2018 at 10:05

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An example of an extra internal law defined from existing internal laws, it's the commutator.

Fix a ring $R$. The commutator of $R$ is the antisymmetric product given by $$[a,b]=ab-ba$$ for two elements $a,b\in R$. One can check that this operation acts as a derivative, meaning that it satisfies Leibniz rule, $$[a,bc]=[a,b]c+b[a,c];$$ and it satisfies the axioms of a Lie algebra product

  • $[a+b,c]=[a,c]+[b,c]$
  • $[a,a]=0$
  • (Jacobi identity) $[a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0$

In this way, $(R,+,[\,\cdot\,,\,\cdot\,])$ becomes a Lie algebra, which is used to understand the commutativity of the original ring $R$.

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The structure $(\Bbb F[x],+,\cdot,\circ)$ is an example of a composition ring.

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