Ternary equivalence relations that are not equivalent to some binary equivalance

Question

1.Is there such a thing as ternary equivalence that is not equivalent or cant not be expressed as binary equivalence?

2.If there is such a thing as expressed in 1, are there any practical uses for it? Is such a an equivalence relation is still transitive? (I know in case of binary equivalence, transitivity is required, but in ternary case is transitivity still a requirement, if yes what does a ternary equivalent relationship might look like )

I am not sure how to phrase this question properly, so if you can understand what is being asked please go ahead and modify this.

Answer 1

Probably the best-known example is the collinearity relation among three points $C(x,y,z)$, which is an equivalence relation in that it satisfies the axioms of...

• Symmetry: $C(x,y,z)$ is invariant under permutations of $x, y, z$
• Reflexivity: $C(x,y,y)$
• Transitivity: $C(a,x,y)\land C(a,y,z)\to C(a,x,z)$

And there doesn't seem to be a reasonable way to "decompose" $C$ in terms of binary relations.