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If, a group is a set, $G$, together with an operation addition, where an operation is a mapping that associates an element of the set to every pair of its elements, satisfying some requirements known as the group axioms.

If, a field is a set $F$ together with two operations called addition and multiplication, where these operations are required to satisfy the field axioms.

What would be the next after field? what are the structures in algebra that defined 3 or even more operations?

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    $\begingroup$ Possibly relevant: mathoverflow.net/questions/120875/… $\endgroup$ – twnly Feb 26 at 6:48
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    $\begingroup$ Well, an additive $2$-category might be a candidate, but that might be overkill. $\endgroup$ – Tobias Kildetoft Feb 26 at 7:55
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    $\begingroup$ @athos en.wikipedia.org/wiki/Algebra_over_a_field $\endgroup$ – James Feb 26 at 7:57
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    $\begingroup$ Hyperstructures They have at least 1 hyperoperations and the output of operations would be sets. I guess. $\endgroup$ – Crunchy Feb 26 at 9:08
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    $\begingroup$ You would have found out at the wiki page examples of structures with even four operations, like Hopf algebras and bialgebras. Maybe you would even consider logical $\vee$ $\wedge$ and $\neg$ as three operations in boolean algebra. $\endgroup$ – rschwieb Feb 26 at 17:56
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There are several algebraic structures with $3$ operations, e.g., a Poisson structure, or a post-Lie algebra structure $(V,[,],\{, \},\cdot)$. The latter has three operations, namely $[x,y]$ and $\{x,y\}$ are Lie brackets on a given vector space $V$ and $x\cdot y$ is a compatible non-associative algebra structure. This comes from differential geometry, quantum field theory, operad theory and many other areas. For an introduction see our paper Affine actions on Lie groups and post-Lie algebra structures.

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