Wondering if there are such things as n-ary algebras, instead of binary. This is in reference to binary operations. All of the algebras listed here revolve around binary operations.
https://en.wikipedia.org/wiki/Outline_of_algebraic_structures#Types_of_algebraic_structures
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In the context of Universal Algebra, an algebra is a structure $\mathbf A = \langle A, F \rangle$, where $A$ is a non-empty set a $F$ is a set of operations on $A$, where by an operation we mean a map $f:A^n \to A$ ($n$ arbitrary); some authors also consider infinitary operations.
So the answer is yes, there are algebras with $n$-ary operations (it's only a matter of defining them, and you can do so arbitrarily).
This might make you ask, then, what is the name of one such algebra? are there some well-known ones?
I don't think there are any such notable algebras, although sometimes we can define them, just to give an example of an algebra with a certain property.
Another reason why these algebras are not so well-known is that, in the finite case (in finite algebras), any $n$-ary operation can be expressed as a composition of binary ones.
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Well, one example is a generalization of the cross product to $(n{-}1)$-ary product over $\mathbb{C}^{n}$ defined as the formal determinant $$ \operatorname{prod}(\vec x_1, \dots, \vec x_{n-1}) = \begin{vmatrix} \vec e_1 & (\vec x_1)_1 & (\vec x_2)_1 & \cdots & (\vec x_{n-1})_1 \\ \vec e_2 & (\vec x_1)_2 & (\vec x_2)_2 & \cdots & (\vec x_{n-1})_2 \\ \vdots & \vdots & \vdots & & \vdots \\ \vec e_{n} & (\vec x_1)_{n} & (\vec x_2)_{n} & \cdots & (\vec x_{n-1})_n \\ \end{vmatrix}, $$ where $\vec e_i$ is the $i$-th coordinate vector.
This is somehow related to the Exterior product. However, as far as I know, this is not that much studied (but I may be wrong).
One example ... in a distributive lattice, it is sometimes convenient to use the ternary "median" operation $$ \mathrm{med}(x,y,z) = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x) = (x \vee y) \wedge (y \vee z) \wedge (z \vee x) $$ In case of a totally ordered set, this is the middle value of the three. But it makes sense more generally (as noted) in a distributive lattice, and satisfies nice identities.