Changes

A comparison of the corresponding projections in <math>\operatorname{Proj}^{(2)} L(\text{A})\!</math> and <math>\operatorname{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\operatorname{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections.  However, to say that a triadic relation <math>L \in \operatorname{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \operatorname{Pow} (O \times S \times I)\!</math> exists such that <math>\operatorname{Proj}^{(2)} L = \operatorname{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\operatorname{Pow} (O \times S \times I).\!</math>

A comparison of the corresponding projections in <math>\operatorname{Proj}^{(2)} L(\text{A})\!</math> and <math>\operatorname{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\operatorname{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections.  However, to say that a triadic relation <math>L \in \operatorname{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \operatorname{Pow} (O \times S \times I)\!</math> exists such that <math>\operatorname{Proj}^{(2)} L = \operatorname{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\operatorname{Pow} (O \times S \times I).\!</math>

As it happens, each of the relations <math>L = L(\text{A})\!</math> or <math>L = L(\text{B})\!</math> is uniquely determined by its projective triple <math>\operatorname{Proj}^{(2)} L.\!</math> This can be seen as follows.

As it happens, each of the relations R C {A, B} is uniquely determined by its projective triple Proj (R).  This can be seen as follows. Consider any coordinate position <s, i> in the plane SxI.  If <s, i> is not in RSI then there can be no element <o, s, i> in R, therefore we may restrict our attention to positions <s, i> in RSI, knowing that there exist at least |RSI| = 8 elements in R, and seeking only to determine what objects o exist such that <o, s, i> is an element in the objective "fiber" of <s, i>. In other words, for what o C O is <o, s, i> C PrSI 1(<s, i>)?  The fact that ROS has exactly one element <o, s> for each coordinate s C S and that ROI has exactly one element <o, i> for each coordinate i C I, plus the "coincidence" of it being the same o at any one choice for <s, i>, tells us that R has just the one element <o, s, i> over each point of SxI.  This proves that both A and B are reducible in an informational sense to triples of dyadic relations, that is, they are "dyadically reducible".

 

Consider any coordinate position <math>(s, i)\!</math> in the plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the objective ''fiber'' of <math>(s, i).\!</math>  In other words, for what <math>o \in O\!</math> is <math>(o, s, i) \in \operatorname{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the &ldquo;coincidence&rdquo; of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.

===6.36. Irreducibly Triadic Relations===

===6.36. Irreducibly Triadic Relations===