n-ary algebras? 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
 A: 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.
A: 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.
A: 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).
