ONT Re: Reductions Among Relations
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RAR. Note 24
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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 recent 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.
Thus, if are looking for a "structure theory" of 2-adic
relations that would identify irreducible primitives
the way that the structure theory of natural numbers
identifies prime numbers as its basis, it will have
to involve other sorts of considerations than just
the relational decomposition of 2-adic relations.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o