By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions.
"Elementary" shall mean "part of the signature of a first-order theory".
The most prominent ternary relation that comes to my mind is the betweenness relation in geometry.
I am looking for examples from all over mathematics of elementary ternary relations that are not binary functions (resp. the corresponding theories).
Examples of elementary quaternary relations are also welcome!
Best Answer
Geometry seems like a natural source, e.g., colinearity of three points. Or how about, for three non-colinear points, clockwise(p,q,r) if the path going through p, q, and then r runs clockwise on the circle whose circumference they lie on? It seems unintuitive to me to try to break up this relation into binary functions and relations.