on Awbrey
&gt; 
&gt; 1.  I agree that if you have some set of binary relations
&gt;     that essentially define the coordinate positions of
&gt;     components in some higher arity relation then there
&gt;     is also the higher arity relation.

You are saying here that the definition of a k-tuple
in terms of its k coordinate projections has nothing
to do with the potential analysis, decomposition, or
whatever, of the whole k-adic relation into possibly
simpler relations, arity-wise.   Welcome to the club!
