It seems that algebraists are particularly interested in sets equipped with binary operations that are well-behaved. I'm curious as to whether mathematicians have studied analogous structures equipped with non-binary operations. For example, why not have a set $G$ with some operation $\phi: G \times G \times G \to G$ such that $G$ is closed with respect to $\phi$? Granted, I'm not sure if the concept of an inverse can be maintained with a non-binary operation, (maybe associativity becomes non-sensical as well...?). Regardless, are there any interesting articles that address these kinds of questions?
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1This is (roughly) getting at operads. – Randall Feb 07 '20 at 13:25
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There are some articles on Wikipedia: Heap (mathematics), Median algebra, Principal homogeneous space. – Calum Gilhooley Feb 07 '20 at 14:22
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This is a FAQ - please see the linked dupe and its links. – Bill Dubuque Feb 07 '20 at 15:45
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https://www.kylem.net/papers/trinary_groups.pdf: This is the kind of thing I was looking for. – Jon Smith Feb 07 '20 at 18:15
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See also Are all n-ary operators simply compositions of binary operators? – Bill Dubuque Feb 07 '20 at 20:05
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@Calum Thanks for catching that. It was what I intended to use but I must have mistakenly touched a neighboring link in the search results. Now fixed. – Bill Dubuque Feb 07 '20 at 20:10
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(Cut-down version of my previous comment.) Another duplicate, already closed: Why does abstract algebra have just binary functions?. – Calum Gilhooley Feb 07 '20 at 20:30