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To what extent do operations and relations overlap? Is there some more general structure that encompasses both of these things?

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Operations or functions are particular cases of relations. For example, consider a set $A$ with a binary operation $+$, that is $+:A\to A$, then one can think of $+$ as a ternary relation $R_+$ on $A$ defined as follows:$$(a,b,c)\in R_+\Longleftrightarrow a+b=c$$ Also this extends to $n$-ary operations as well, any $n$-ary operation is nothing but a $n+1$-ary relation.

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