ONT Re: Reductions Among Relations
RAR. Note 4
Projective Reduction of Triadic Relations (cont.)
There are a number of preliminary matters that
will need to be addressed before I can proceed.
Last time I gave two cases of 3-adic (or triadic) relations
with projective reductions to 2-adic (or dyadic) relations,
by name, "triadics reducible over projections" (TROP's) or
"triadics reconstructible out of projections" (TROOP's).
Still, one needs to be very careful and hedgey about saying,
even in the presence of such cases, that "all is dyadicity".
I will make some attempt to explain why in the next episode,
and then I will take up examples of 3-adics that happen to
be irreducible in this sense, in effect, that are not able
to be recovered uniquely from their 2-adic projection data.
Call them "triadics irreducible over projections" (TRIOP's)
or "projectively irreducible triadics" (PIT's).
In the story of A and B, it appears to be the case
that that the triadic relations L(A) and L(B) are
distinguished from each other, and what's more,
distinguished from all of the other relations
in the garden of OSI, for the same O, S, I.
At least, so says I and my purported proof.
I am so suspicious of this result myself that
I will probably not really believe it for a while,
until I have revisited the problem and the "proof"
a few times, to see if I can punch any holes in it.
But let it pass for proven for now,
and let my feeble faith go for now.
For the sake of a more balanced account,
it's time to see if we can dig up any cases
of "projectively irreducible triadics" (PIT's).
Any such PIT relation, should we ever fall into one,
is bound to occasion another, since it is a porismatic
part of the definition that a 3-adic relation L is a PIT
if and only if there exists a distinct 3-adic relation L'
such that the 2-adic faces of L and L' are indiscernible.
In this event, then both L and L' fall into the de-genre
of PIT's together.
Well, PIT's are not far to find, once you think to look for them --
indeed, the landscape of "formal or mathematical existence" (FOME)
is both figuratively and litterally rife with them!
What follows is the account of a couple,
that I will dub "L_0" and "L_1".
But first, even though the question of projective reduction
has to do with 3-adic relations as a general class, and is
thus independent of their potential use as sign relations,
it behooves us to consider the bearing of these reduction
properties on the topics of interest to us for the sake
of communication and representation via sign relations.
| Nota Bene. On the Variety and Reading of Subset Notations.
| Let any of the locutions, L c XxYxZ, L on XxYxZ, L sub XxYxZ,
| substitute for the peculiar style of "in-line" or "in-passing"
| reference to subsethood that has become idiomatic in mathematics,
| and that would otherwise use the symbol that has been customary
| since the time of Peano to denote "contained in" or "subset of".
Most likely, any triadic relation L on XxYxZ that is imposed on
the arbitrary domains X, Y, Z could find use as a sign relation,
provided that it embodies any constraint at all, in other words,
so long as it forms a proper subset L of the entire space XxYxZ.
But these sorts of uses of triadic relations are not guaranteed
to capture or constitute any natural examples of sign relations.