ONT Re: Relations And Their Divisitudes
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RATD. Note 33
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Prospects for the Compositional Analysis of Relations
There are a number of very instructive observations that we might make
at this point. One of the most striking is that a composite relation
can be a very simple sort of relation, for all its being compounded
of other relations. Indeed, in our earlier example, G o H is the
elementary relation 4:4, and yet it is evidently composed of the
2-adic relations G and H. What's more, there is nothing unique
about this decomposition, as many other pairs of factors would
be capable of producing the same result. What this tells us
is that the complexity of a 2-adic relation is not strongly
related to its properties under relational decomposition.
Consequently, if we are seeking a "structure theory" of 2-adic relations
that is capable of identifying irreducible primitives in something like
the same way that the structure theory of natural numbers identifies
prime numbers as its basis, then it will necessarily involve other
sorts of considerations about the 2-adic relations being analyzed
than just their relational decompositions, that is, the pairings
of 2-adic relations from which they are conceivably composed.
Jon Awbrey
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