— A little memo —

While I was browsing through an abstract of a recent paper, which is about something called “hypergraph labeling”, I was brought back to my old memory.

Quite some time ago, I happened to skim through a fragment written by Charles Sanders Peirce, which goes something like “ternary relation cannot be reduced to a set of binary releations”, and yet “n-ary relation, where n > 3, can be reduced to a set of ternary relations”.  And he gave an example to show his point.  If my memory serves well, the example was kinda like this:

Let’s take a ternary relation “A gives B to C”.
Even if binary relations “A parts with B”, “B is received by A”… and so on are satisfied simultaneously, they do not compose the ternary relation “A gives B to C”.

Interesting a bit.  His argument was somehow convincing to me but not entirely.  I had been caught by this argument for some time and sought some “nicer” demonstration for the proposition, even though I was not clear at all about what I really meant by “nicer”.

Graph can represent a set of binary relations, but not n-ary relations (n > 2), whereas hypergraph can.  Maybe something like this was what Peirce had in his mind?

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