Operations of arity 3 naturally arise in universal algebra. For example, one strand of research is to characterize the properties of the lattice of congruences of a variety by the existence of special terms -- these usually have arity 3. For example, if a variety has a ternary operation m(x, y, z) such that m(x, y, y) = x and m(x, x, y) = y, then the lattice of congruences is modular. (The converse is not true, but there is a weaker statement involving ternary operations that is true.) Examples of this include groups ($m(x, y, z) = x y^{-1} z$) and vector spaces ($m(x, y, z) = x - y + z$).

The ternary operation for vector spaces has a natural geometric interpretation as vector addition in affine space, where vectors are not required to be based at the origin. If you draw a vector from y to x and a vector from y to z, then $m(x, y, z)$ is the vector from y to x + z. You can think of addition as defined by drawing a parallelogram $xyzw$. Then $m(x,y,z)=w$.