Interesting properties of ternary relations?

Question

Many people are familiar with some properties of binary relations, such as reflexivity, symmetry and transitivity.

What are the commonly studied properties of ternary (3-ary) relations?

If you could provide a motivating example of why the property is interesting that would also be helpful.

Answer 1

One interesting kind of ternary relation is the "betweenness" relation characterised by the Axioms of Order in Hilbert's Foundations of Geometry.

I expect ternary relations are typically studied less because identifying an interesting one requires much more involved definitions than is normally the case for binary relations...

Answer 2

One interesting example is "being Steiner triple system" (and this is has a connection with Qiaochu Yuan's comment: any Steiner triple system defines commutative quasigroup).

Answer 3

Pythagorean triples induce a ternary relation that has many interesting properties.

Answer 4

One class of example arises in Lie theory. Take $L$ a simple Lie algebra. Then there is a particular $\mathfrak{sl}(2)\subset L$, name take $E$ to be a highest root, $F$ a lowest root, and $H=[E,F]$. Then decompose $L$ as a representation of this subalgebra. You get $L=L_0\oplus L_1\otimes T \oplus \mathfrak{sl}(2)$. Then $L_1$ is a ternary system. This satisfies a (complicated) identity. You can reconstruct $L$ from the ternary system and you need this identity for the Jacobi identity.

Answer 5

So far nobody is actually giving properties, but just examples. I'll continue that theme.

When Gauss defined composition of quadratic forms, on the level of quadratic forms what he defined was not really a law of composition but a ternary relation (three quadratic forms $Q_1$, $Q_2$, and $Q_3$ are "in composition" if $Q_1(x,y)Q_2(x',y') = Q_3(B,B')$ where $B$ and $B'$ are linear in $xx', xy', yx', yy'$). At the level of proper equivalence classes of quadratic forms this becomes a group law.

You could say any group law is defined by a ternary relation $ghk = 1$ on the group. This fits the geometric description and addition of points on an elliptic curve or Bhargava's interpretation of Gauss's composition.