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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)

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→‎6.35. Reducibility of Sign Relations: markup

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| <math>\operatorname{Proj}^{(2)} : \operatorname{Pow}(X \times Y \times Z) \to \operatorname{Explo}(X, Y, Z ~|~ 2).\!</math>

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In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\operatorname{Proj}^{(2)}\!</math> is injective. The mapping <math>\operatorname{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \operatorname{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation <math>L \in \operatorname{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\operatorname{Proj}^{(2)})^{-1}(\operatorname{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\operatorname{Proj}^{(2)}L = \operatorname{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''. Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\operatorname{Proj}^{(2)}.\!</math>

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In this setting, the issue of whether triadic relations are "reducible to" or "reconstructible from" their dyadic projections, both in general and in specific cases, can be identified with the question of whether Proj is injective. The mapping Proj : Pow (XxYxZ) > Explo (X, Y, Z; 2) is said to "preserve information" about the triadic relations R C Pow (XxYxZ) if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation R C Pow (XxYxZ), it can be said that R is "determined by" ("reducible to" or "reconstructible from") its dyadic projections if and only if Proj 1(Proj (R)) is the singleton set {R}. Otherwise, there exists an R' such that Proj (R) = Proj (R'), and in this case R is said to be "irreducibly triadic" or "genuinely triadic". Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under Proj.

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The next series of Tables illustrates the operation of Proj by means of its actions on the sign relations A and B. For ease of reference, Tables 69.1 and 70.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising Proj (A) and Proj (B) are shown in Tables 69.2 to 69.4 and Tables 70.2 to 70.4, respectively.

Jon Awbrey

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