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 . By this I mean,
Do there exist two binary relations and , both closed in , so that
For a certain , if there exist two ordered pairs and such that ,
Then we have a possibility that, . This is just a a possibility, it need not hold. This makes the cardinality of significant.
If is finite, for to be closed in , the co-domain of should not have more elements than in . We need to pack elements in boxes, bringing (2) into play. This partitions the set of ordered pairs. Different maps count the number of ordered pairs of binary relations satisfying (1).
If is infinite, the do not encounter some of the problems as has same cardinality as . Further, every complete order of (every partial order is extendable to a complete order) gives rise to an natural ordering of , thus giving rise to a canonical bijection.
Some simple restrictions on shall drastically affect the validality of (1). Like suppose, is permutative, that is for some permutation . 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 ? That is, when do we have satisfying
2. Can we have restrictions on which render the impossibility of ? For smaller finite , 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 … ?