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The next few sections deal with the informational relationships that exist between <math>n\!</math>-place relations and the relations of fewer dimensions that arise as their projections. A number of set-theoretic constructions of constant use in this investigation are brought together and described in the present section. Because their intended application is mainly to sign relations and other triadic relations, and since the current focus is restricted to discrete examples of these types, no attempt is made to present these constructions in their most general and elegant fashions, but only to deck them out in the forms that are most readily pressed into immediate service.
The next few sections deal with the informational relationships that exist between <math>n\!</math>-place relations and the relations of fewer dimensions that arise as their projections. A number of set-theoretic constructions of constant use in this investigation are brought together and described in the present section. Because their intended application is mainly to sign relations and other triadic relations, and since the current focus is restricted to discrete examples of these types, no attempt is made to present these constructions in their most general and elegant fashions, but only to deck them out in the forms that are most readily pressed into immediate service.
An initial set of operations, required to establish the subsequent constructions, all have in common the property that they do exactly the opposite of what is normally done in abstracting sets from situations. These operations reconstitute, though still in a generic, schematic, or stereotypical manner, some of the details of concrete context and interpretive nuance that are commonly suppressed in forming sets. Stretching points back along the direction of their initial pointing out, these extensions return to the mix a well-chosen selection of features, putting back in those dimensions from which ordinary sets are forced to abstract and in their ordination to treat as distractions.
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In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, "clipboards" and "scrapbooks":
In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, "clipboards" and "scrapbooks":
