Ternary relation

Here is an interesting discussion I had with a friend of mine. Rather i must say, it is one amongst the many interesting discussions I have had someone who more than just a friend.

He said: “We can study the limitation of human thought by computers” … he meant … “Computers do kind of reflect the limitation of human thought process”.

Umm … Philosophical, I thought.

One of the things he said elaborating his idea was as follows: “A ternary relation (or any n-ary relation) is always expressible as some interaction of binary relations”.

One (counter?)example which we are not able to still decide about whether the map from a set of three points not all on a line to their orthrocentre defines one.

I approached it as follows. Let $*$ be a ternary relation, closed in set $S$. By this I mean,

$\forall \, x,y,z \in S,\quad *(x,y,z)=w\in S$

Do there exist two binary relations $\phi$ and $\varphi$, both closed in $S$, so that

$\forall \, x,y,z \in S,\quad*(x,y,z)=\phi(x,\varphi (y,z))\qquad \qquad \dots (1)$

For a certain $x\in S$, if there exist two ordered pairs $(y,z)$ and $(y_1,z_1)$ such that $(y,z)\ne (y_1,z_1)$,

$(x,y,z)=(x,y_1,z_1)\qquad \qquad \dots (2)$

Then we have a possibility that, $\varphi(y,z)=\varphi(y_1,z_1)$. This is just a a possibility, it need not hold. This makes the cardinality of $S$ significant.

If $S$ is finite, for $\varphi$ to be closed in $S$, the co-domain of $\varphi$ should not have more elements than in $S$. We need to pack $2{|S|\choose 2}$ elements in $n$ boxes, bringing (2) into play. This partitions the set of ordered pairs. Different maps $n^{2{|S|\choose 2}}$ count the number of ordered pairs of binary relations $(\phi,\varphi)$ satisfying (1).

If $S$ is infinite, the do not encounter some of the problems as $S\times S$ has same cardinality as $S$. Further, every complete order of $S$ (every partial order is extendable to a complete order) gives rise to an natural ordering of $S\times S$, thus giving rise to a canonical bijection.

Some simple restrictions on $*$ shall drastically affect the validality of (1). Like suppose, $*$ is permutative, that is $*(x,y,z)=*(\delta (x), \delta (y), \delta (z))$ for some permutation $\delta$. Further finding such relations makes me clueless.

Challenges we have:

1. What is the general solution (in terms of mappings or solutions) when $*$ is n-ary for some finite $n$? That is, when do we have $\phi_1,\phi_2,\dots, \phi_{n-1}$ satisfying

$*(x_1,x_2,\dots, x_n)=\phi_1(x_1,\phi_2(x_2\dots \phi_{n-1}(x_{n-1},x_{n}\underbrace{)\dots)}_{n-1}$

2. Can we have restrictions on $*$ which render the impossibility of $(1)$? For smaller finite $n$, this is computer doable.

3. The composition construct may be a goof up, there could be better ways of achieving a ternary relation in terms of binary relations.

Far in the horizons … ?