Relation reduction

The 3-adic relations ${\displaystyle L_{0}\!}$ and ${\displaystyle L_{1}\!}$ are shown in the next two Tables:

${\displaystyle L_{0}~=~\{(x,y,z)\in \mathbb {B} ^{3}:x+y+z=0\}\!}$

${\displaystyle X\!}$	${\displaystyle Y\!}$	${\displaystyle Z\!}$
${\displaystyle 0\!}$	${\displaystyle 0\!}$	${\displaystyle 0\!}$
${\displaystyle 0\!}$	${\displaystyle 1\!}$	${\displaystyle 1\!}$
${\displaystyle 1\!}$	${\displaystyle 0\!}$	${\displaystyle 1\!}$
${\displaystyle 1\!}$	${\displaystyle 1\!}$	${\displaystyle 0\!}$

${\displaystyle L_{1}~=~\{(x,y,z)\in \mathbb {B} ^{3}:x+y+z=1\}\!}$

${\displaystyle X\!}$	${\displaystyle Y\!}$	${\displaystyle Z\!}$
${\displaystyle 0\!}$	${\displaystyle 0\!}$	${\displaystyle 1\!}$
${\displaystyle 0\!}$	${\displaystyle 1\!}$	${\displaystyle 0\!}$
${\displaystyle 1\!}$	${\displaystyle 0\!}$	${\displaystyle 0\!}$
${\displaystyle 1\!}$	${\displaystyle 1\!}$	${\displaystyle 1\!}$

A 2-adic projection of a 3-adic relation ${\displaystyle L\!}$ is the 2-adic relation that results from deleting one column of the table for ${\displaystyle L\!}$ and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored.

In the case of the above two relations, ${\displaystyle {L_{0},L_{1}~\subseteq ~X\times Y\times Z~\cong ~\mathbb {B} ^{3}},\!}$ the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

${\displaystyle \mathrm {proj} _{XY}L_{0}\!}$ ${\displaystyle X\!}$ ${\displaystyle Y\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$

${\displaystyle \mathrm {proj} _{XZ}L_{0}\!}$ ${\displaystyle X\!}$ ${\displaystyle Z\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$

${\displaystyle \mathrm {proj} _{YZ}L_{0}\!}$ ${\displaystyle Y\!}$ ${\displaystyle Z\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$

${\displaystyle \mathrm {proj} _{XY}L_{1}\!}$ ${\displaystyle X\!}$ ${\displaystyle Y\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$

${\displaystyle \mathrm {proj} _{XZ}L_{1}\!}$ ${\displaystyle X\!}$ ${\displaystyle Z\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$

${\displaystyle \mathrm {proj} _{YZ}L_{1}\!}$ ${\displaystyle Y\!}$ ${\displaystyle Z\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$

It is clear on inspection that the following three equations hold:

${\displaystyle \mathrm {proj} _{XY}(L_{0})~=~\mathrm {proj} _{XY}(L_{1})~\!}$

${\displaystyle \mathrm {proj} _{XZ}(L_{0})~=~\mathrm {proj} _{XZ}(L_{1})~\!}$

${\displaystyle \mathrm {proj} _{YZ}(L_{0})~=~\mathrm {proj} _{YZ}(L_{1})~\!}$

These equations say that ${\displaystyle L_{0}\!}$ and ${\displaystyle L_{1}\!}$ cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, each relation is said to be irreducible with respect to 2-adic projections. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is projectively irreducible, the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations.

The 3-adic relations ${\displaystyle L_{\mathrm {A} }\!}$ and ${\displaystyle L_{\mathrm {B} }\!}$ are shown in the next two Tables:

${\displaystyle L_{\mathrm {A} }~=~{\text{Sign Relation of Interpreter A}}\!}$

${\displaystyle {\text{Object}}\!}$	${\displaystyle {\text{Sign}}\!}$	${\displaystyle {\text{Interpretant}}\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$

${\displaystyle L_{\mathrm {B} }~=~{\text{Sign Relation of Interpreter B}}\!}$

${\displaystyle {\text{Object}}\!}$	${\displaystyle {\text{Sign}}\!}$	${\displaystyle {\text{Interpretant}}\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {A} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$
${\displaystyle \mathrm {B} \!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$	${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$

In the case of the two sign relations, ${\displaystyle L_{\mathrm {A} },L_{\mathrm {B} }~\subseteq ~X\times Y\times Z~\cong ~O\times S\times I,\!}$ the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.

${\displaystyle \mathrm {proj} _{XY}(L_{\mathrm {A} })\!}$ ${\displaystyle {\text{Object}}\!}$ ${\displaystyle {\text{Sign}}\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$

${\displaystyle \mathrm {proj} _{XZ}(L_{\mathrm {A} })\!}$ ${\displaystyle {\text{Object}}\!}$ ${\displaystyle {\text{Interpretant}}\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$

${\displaystyle \mathrm {proj} _{YZ}(L_{\mathrm {A} })\!}$ ${\displaystyle {\text{Sign}}\!}$ ${\displaystyle {\text{Interpretant}}\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$

${\displaystyle \mathrm {proj} _{XY}(L_{\mathrm {B} })\!}$ ${\displaystyle {\text{Object}}\!}$ ${\displaystyle {\text{Sign}}\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$

${\displaystyle \mathrm {proj} _{XZ}(L_{\mathrm {B} })\!}$ ${\displaystyle {\text{Object}}\!}$ ${\displaystyle {\text{Interpretant}}\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {A} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle \mathrm {B} \!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$

${\displaystyle \mathrm {proj} _{YZ}(L_{\mathrm {B} })\!}$ ${\displaystyle {\text{Sign}}\!}$ ${\displaystyle {\text{Interpretant}}\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {A} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {u} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {B} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$ ${\displaystyle {}^{\backprime \backprime }\mathrm {i} {}^{\prime \prime }\!}$

It is clear on inspection that the following three inequalities hold:

${\displaystyle \mathrm {proj} _{XY}(L_{\mathrm {A} })~\neq ~\mathrm {proj} _{XY}(L_{\mathrm {B} })\!}$

${\displaystyle \mathrm {proj} _{XZ}(L_{\mathrm {A} })~\neq ~\mathrm {proj} _{XZ}(L_{\mathrm {B} })\!}$

${\displaystyle \mathrm {proj} _{YZ}(L_{\mathrm {A} })~\neq ~\mathrm {proj} _{YZ}(L_{\mathrm {B} })\!}$

These inequalities say that ${\displaystyle L_{\mathrm {A} }\!}$ and ${\displaystyle L_{\mathrm {B} }\!}$ can be distinguished from each other solely on the basis of their 2-adic projection data. But this is not enough to say that either one of them is projectively reducible to their 2-adic projection data. To say that a 3-adic relation is projectively reducible in that respect, one has to show that it can be distinguished from every other 3-adic relation on the basis of the 2-adic projection data alone.

In other words, to show that a 3-adic relation ${\displaystyle L\!}$ on ${\displaystyle O\times S\times I\!}$ is reducible or reconstructible in the 2-adic projective sense, it is necessary to show that no distinct ${\displaystyle L'\!}$ on ${\displaystyle O\times S\times I\!}$ exists such that ${\displaystyle L\!}$ and ${\displaystyle L'\!}$ have the same set of projections. Proving this takes a much more comprehensive or exhaustive investigation of the space of possible relations on ${\displaystyle O\times S\times I\!}$ than looking merely at one or two relations at a time.

Fact. As it happens, each of the relations ${\displaystyle L_{\mathrm {A} }\!}$ and ${\displaystyle L_{\mathrm {B} }\!}$ is uniquely determined by its 2-adic projections. This can be seen by following the proof that is given below.

Before tackling the proof, however, it will speed things along to recall a few ideas and notations from other articles.

• If ${\displaystyle L\!}$ is a relation over a set of domains that includes the domains ${\displaystyle U\!}$ and ${\displaystyle V,\!}$ then the abbreviated notation ${\displaystyle L_{UV}\!}$ can be used for the projection ${\displaystyle \mathrm {proj} _{UV}(L).\!}$

• The operation of reversing a projection asks what elements of a bigger space project onto given elements of a smaller space. The set of elements that project onto ${\displaystyle x\!}$ under a given projection ${\displaystyle f\!}$ is called the fiber of ${\displaystyle x\!}$ under ${\displaystyle f\!}$ and is written ${\displaystyle f^{-1}(x)\!}$ or ${\displaystyle f^{-1}x.\!}$

• If ${\displaystyle X\!}$ is a finite set, the cardinality of ${\displaystyle X,\!}$ written ${\displaystyle \mathrm {card} (X)\!}$ or ${\displaystyle |X|,\!}$ means the number of elements in ${\displaystyle X.\!}$

Proof. Let ${\displaystyle L\!}$ be either one of the relations ${\displaystyle L_{\mathrm {A} }\!}$ or ${\displaystyle L_{\mathrm {B} }.\!}$ Consider any coordinate position ${\displaystyle (s,i)\!}$ in the ${\displaystyle SI\!}$-plane ${\displaystyle S\times I.\!}$ If ${\displaystyle (s,i)\!}$ is not in ${\displaystyle L_{SI}\!}$ then there can be no element ${\displaystyle (o,s,i)\!}$ in ${\displaystyle L,\!}$ therefore we may restrict our attention to positions ${\displaystyle (s,i)\!}$ in ${\displaystyle L_{SI},\!}$ knowing that there exist at least ${\displaystyle |L_{SI}|=8\!}$ elements in ${\displaystyle L,\!}$ and seeking only to determine what objects ${\displaystyle o\!}$ exist such that ${\displaystyle (o,s,i)\!}$ is an element in the fiber of ${\displaystyle (s,i).\!}$ In other words, for what ${\displaystyle o\!}$ in ${\displaystyle O\!}$ is ${\displaystyle (o,s,i)\!}$ in the fiber ${\displaystyle \mathrm {proj} _{SI}^{-1}(s,i)?\!}$ Now, the circumstance that ${\displaystyle L_{OS}\!}$ has exactly one element ${\displaystyle (o,s)\!}$ for each coordinate ${\displaystyle s\!}$ in ${\displaystyle S\!}$ and that ${\displaystyle L_{OI}\!}$ has exactly one element ${\displaystyle (o,i)\!}$ for each coordinate ${\displaystyle i\!}$ in ${\displaystyle I,\!}$ plus the “coincidence” of it being the same ${\displaystyle o\!}$ at any one choice for ${\displaystyle (s,i),\!}$ tells us that ${\displaystyle L\!}$ has just the one element ${\displaystyle (o,s,i)\!}$ over each point of ${\displaystyle S\times I.\!}$ All together, this proves that both ${\displaystyle L_{\mathrm {A} }\!}$ and ${\displaystyle L_{\mathrm {B} }\!}$ are reducible in an informative sense to 3-tuples of 2-adic relations, that is, they are projectively 2-adically reducible.

The projective analysis of 3-adic relations, illustrated by means of concrete examples, has been pursued just far enough at this point to state this clearly demonstrated result:

• Some 3-adic relations are, and other 3-adic relations are not, reducible to, or reconstructible from, their 2-adic projection data. In short, some 3-adic relations are projectively reducible and some 3-adic relations are projectively irreducible.

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