Why does abstract algebra have just binary functions?

Question

Are there operations/functions that take any other than 2 arguments in abstract algebra? If there are, then why are they not used or shown while teaching the topic? If there are not, then why is the subject constrained to just binary functions?

Answer 1

Inversion is a unary operation. And the cross-product, which is a binary operation, can be generalized to a $n$-ary operation from $\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\times\cdots\times\mathbb{R}^{n+1}$ ($n$ times) into $\mathbb R^{n+1}$.

Answer 2

Aside from the usual nullary (constants), unary operations, and binary operations, there are ternary operations also. One example is that of Heaps and semi-heaps which are intended to axiomatize operations such as $\, a,b,c \to ab^{-1}c\,$ in groups. The intent of this operation is analogous to the difference between vector spaces and affine spaces. In a vector space there is a distinguished zero vector, while in an affine space there isn't. Also, in an affine space there is a multi-variable operation of the affine combination of points.