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===6.36. Irreducibly Triadic Relations===

===6.36. Irreducibly Triadic Relations===

Most likely, any triadic relation <math>L \subseteq X \times Y \times Z\!</math> imposed on arbitrary domains <math>X, Y, Z\!</math> could find use as a sign relation, provided it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, a relationship symbolized by writing <math>L \subset X \times Y \times Z.\!</math> However, triadic relations of this sort are not guaranteed to form the most natural examples of sign relations.

Most likely, any triadic relation R c XxYxZ that is imposed on 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 of its total space, R c XxYxZ.  However, these sorts of uses of triadic relations are not guaranteed to capture the most natural examples of sign relations.

In order to show what an irreducibly triadic relation looks like, this section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone.  As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.

In order to show what an irreducibly triadic relation looks like, this section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone.  As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.

Tables 71.1 and 72.1 show a pair of irreducibly triadic relations R0 and R1, respectively.  Tables 71.2 to 71.4 and Tables 72.2 to 72.4 show the dyadic relations comprising Proj (R0) and Proj (R1), respectively.

Tables 71.1 and 72.1 show a pair of irreducibly triadic relations R0 and R1, respectively.  Tables 71.2 to 71.4 and Tables 72.2 to 72.4 show the dyadic relations comprising Proj (R0) and Proj (R1), respectively.