Inversion in the sense of negation? But why focus on 2-ary inversion, and not 3-ary inversion (where a, wa, and w^2a are considered a triple of 3-ary inverses, for w a primitive 3rd root of unity), or whatever n-ary thing. Perhaps this is circular too…
Once you focus on the relations that are like functions in that they satisfy conditions like "for any value of _ there is unqiue _", there is something 2-ish going on, because the places in the relation are being partitioned into 2 pieces, the domain piece and the codomain piece.
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Even if you try to generalize bijections by using a condition that the relation must be function-like for multiple domain/codomain partitions (e.g. for any values of any k places, there are designated values for other n-k), each partition is into 2 parts.
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A ternary correspondence is a subset of X x Y x Z such that each element of X, Y, or Z appears in precisely one element of the subset.
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You can alternatively phrase this in 2-ish ways if you like, but you can phrase anything in lots of ways. It's not fundamentally a 2-ish notion, except in that each 3-point simplex contains within it 2-simplices for its sides, and so on.
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As I said, you can express all the higher n-ary structure from 2-ary structure, yes, but this needn't be considered the primary presentation.
End of conversation
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