Changes

This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.

This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.

By means of these constructions the operation that forms <math>\operatorname{Proj}^{(2)} L\!</math> for each triadic relation <math>L \subseteq X \times Y \times Z\!</math> can be expressed as a function:

By means of these constructions the operation that forms Proj (R) for each triadic relation R c XxYxZ can be expressed as a function:

Proj : Pow (XxYxZ) > Explo (X, Y, Z; 2).

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

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.

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.