SUO: Re: Reductions Among Relations
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
RARest Of All Interest Groups:
This will complete the revision of this RARified thread from last Autumn.
I will wind it up, as far as this part of it goes, by recapitulating the
development of the "Rise" relation, from a couple of days ago, this time
working through its analysis and its synthesis as fully as I know how at
the present state of my knowledge. The good of this exercise, of course,
the reason for doing all of this necessary work, is not because the Rise
relation is so terribly interesting in itself, but rather to demonstrate
the utility of the functional framework and its sundry attached tools in
their application to a nigh unto minimal and thus least obstructive case.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
The good of this whole discussion, the use of it all,
the thing about it that makes it worth going through,
at least, for me, is not just to settle doubts about
the "banal", "common", or figuratively and literally
"trivial" (Latin for locus where "three roads" meet)
type of issue that may have appeared to be its point,
but, as I said in my recent reprise of justification,
to examine and explore "the extent to which it is possible to
construct relations between complex relations and simpler
relations. The aim here, once we get past questions of
what is reducible in what way and what not in no way,
is to develop concrete and fairly general methods
for analyzing the structures of those relations
that are indeed amenable to a useful analysis --
and here I probably ought to emphasize that
I am talking about the structure of each
relation in itself, at least, to the
extent that it presents itself in
extensional form, and not just
the syntax of this or that
relational expression."
So let me run through this development once more,
this time interlacing its crescendoes with a few
supplemental notes of showcasing or sidelighting,
aimed to render more clearly the aim of the work.
In order to accumulate a stock of ready-mixed concrete instances,
at the same time to supply ourselves with relatively fundamental
materials for building ever more complex and prospectively still
more desirable and elegant structures, maybe, even if it must be
worked out just a little bit gradually, hopefully, incrementally,
and even at times jury-rigged here and there, increasingly still
more useful architectronic forms for our joint contemplation and
and our meet edification, let us then set out once more from the
grounds of what we currently have in our command, and advance in
the directions of greater generalities and expanded scopes, with
the understanding that many such journeys are possible, and that
each is bound to open up on open-ended views at its unlidded top.
By way of a lightly diverting overture, let's begin
with an exemplar of a "degenerate triadic relation" --
do you guess that our opera is in an Italian manor? --
a particular version of the "between relation", but
let us make it as simple as we possibly can and not
attempt to analyze even that much of a case in full
or final detail, but leave something for the finale.
Let B = {0, 1}.
Let the relation named "Rise<2>"
such that Rise<2> c B^2 = B x B,
be exactly this set of 2-tuples:
| Rise<2> = {<0, 0>,
| <0, 1>,
| <1, 1>}
Let the relation named "Rise<3>"
such that Rise<3> c B^3 = BxBxB,
be exactly this set of 3-tuples:
| Rise<3> = {<0, 0, 0>,
| <0, 0, 1>,
| <0, 1, 1>,
| <1, 1, 1>}
Then Rise<3> is a "degenerate 3-adic relation"
because it can be expressed as the conjunction
of a couple of 2-adic relations, specifically:
Rise<3><x, y, z> <=> [Rise<2><x, y> and Rise<2><y, z>].
But wait just a minute! You read me clearly to say already --
and I know that you believed me! -- that no 3-adic relation
can be decomposed into any 2-adic relations, so what in the
heck is going on!? Well, "decomposed" implies the converse
of "composition", which has to mean "relational composition"
in the present context, and this composition is a different
operation entirely from the "conjunction" that was employed
above, to express Rise<3> as a conjunction of two Rise<2>'s.
That much we have seen and done before, but in the spirit of
that old saw that "what goes up must come down" we recognize
that there must be a supplementary relation in the scheme of
things that is equally worthy of our note, and so let us dub
this diminuendo the "Fall" relation, and set to define it so:
Let the relation named "Fall<2>"
such that Fall<2> c B^2 = B x B,
be exactly this set of 2-tuples:
| Fall<2> = {<0, 0>,
| <1, 0>,
| <1, 1>}
Let the relation named "Fall<3>"
such that Fall<3> c B^3 = BxBxB,
be exactly this set of 3-tuples:
| Fall<3> = {<0, 0, 0>,
| <1, 0, 0>,
| <1, 1, 0>,
| <1, 1, 1>}
And on those notes I must rest ...
To be continued ...
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤