# Interesting properties of ternary relations?

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.

• Whether they describe a function S x S -> S. Whether that function is associative. Whether it is commutative... – Qiaochu Yuan Aug 3 '10 at 5:43
• I'm now sure why you put that as a comment rather than as an answer. – bryn Aug 3 '10 at 7:20
• community wiki, probably? – Grigory M Aug 3 '10 at 8:39
• @Grigory: This one is hard to call – Casebash Aug 3 '10 at 8:50
• Some references that may be of interest. On page 146 of dml.cz/bitstream/handle/10338.dmlcz/128381/… there is a list of properties of ternary relations. References  and  of that paper discuss other properties. – Doug Chatham Aug 3 '10 at 14:26

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...

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).

Many ternary properties are interesting, and many of them are learned in school before properties of binary relations. I define any ternary relation as a relation that is expressed as a three-place predicate. For example, consider the three place predicate, '_____added to_____yields_____'. This particular 3-adic relation has a number of interesting properties, including:

the commutative property: $$\forall x \forall y \forall z : R(x,y,z) \implies R(y,x,z)$$

the associative property: $$\forall w \forall x \forall y \forall z : R(s(w,x),y,z) \implies R(w,s(x,y),z)$$, where the function $$s$$ takes the sum of its inputs.

et cetera.

Note that these two properties are also properties of the relation '_____multiplied by_____yields_____'.

These two ternary relations differ, however, on this property (the additive identity property): $$\forall x : R(x,0,x)$$

For all quantities, $$x$$ added to $$0$$ yields $$x$$, but it is not the case that for all quantities $$x$$, $$x$$ multiplied by $$0$$ yields $$x$$.

Notice that the 3-adic relation '_____and_____sit to either side of_____on the sofa' also shares the commutative property.

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.

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.

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

• I'd be interested to hear some of those properties. The only one I can think of (off the top of my head) is R(x,y,z) <-> R(y,x,z). – bryn Aug 7 '10 at 7:58
• See Alperin's "The modular tree of Pythagoras", cited in the wikipedia article. It shows that all triples can be obtained from (3,4,5) by simple matrix operations. – lhf Aug 7 '10 at 11:11
• Okay. That is interesting but it's not the type of property that I was after -- I want properties expressible in first order logic (+ identity). – bryn Aug 12 '10 at 9:20
• @bryn: see my post here for more on the ternary tree of primitive Pythagorean triples – Bill Dubuque Oct 18 '10 at 21:52