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===6.34. Set Theoretic Constructions===
===6.34. Set-Theoretic Constructions===
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The next few sections deal with the informational relationships that exist between n 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.
In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, "clipboards" and "scrapbooks":
1. The smaller and shorter term index sets, typically having the form I = {1, ... , n}, are used to keep tabs on the terms of finite sets and sequences, unions and intersections, sums and products.
In this context and elsewhere, the notation [n] = {1, ... , n} will be used to refer to a "standard segment" (finite initial subset) of the natural numbers N = {1, 2, 3, ... }.
2. The larger and longer term index sets, typically having the form J c N = {1, 2, 3, ... }, are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.
Definition. An "indicated set" j^S is an ordered pair j^S = <j, S>, where j C J is the indicator of the set and S is the set indicated.
Definition. An "indited set" j^S extends the incidental and extraneous indication of a set into a constant indictment of its entire membership.
j^S = j^{j^s : s C S} = j^{<j, s> : s C S} = <j, {j} x S>.
Notice the difference between these notions and the more familiar concepts of an "indexed set", "numbered set", and "enumerated set". In each of these cases the construct that results is one where each element has a distinctive index attached to it. In contrast, the above indications and indictments attach to the set S as a whole, and respectively to each element of it, the same index number j.
Definition. An "indexed set" <S, L> is constructed from two components: its "underlying set" S and its "indexing relation" L : S > N, where L is total at S and tubular at N. It is defined as follows:
<S, L> = { {s} x L(s) : s C S } = {<s, j> : s C S, j C L(s)}.
L assigns a unique set of "local habitations" L(s) to each element s in the underlying set S.
Definition. A "numbered set" <S, f>, based on the set S and the injective function f : S > N, is defined as follows. ???
Definition. An "enumerated set" <S, f> is a numbered set with a bijective f. ???
The "n fold sum" ("co product", "disjoint union") of the sets X1, ... , Xn is notated and defined as follows:
Ui Xi = X1 + ... + Xn = 1^X1 U ... U n^Xn.
The "n fold product" ("cartesian product") of the sets X1, ... , Xn is notated and defined as follows:
Xi Xi = X1 x ... x Xn = {<x1, ... , xn> : xi C Xi for all i}.
As an alternative definition, the n tuples of Xi Xi can be regarded as sequences of elements that come from the successive Xi, and thus as the various functions of a certain sort that map [n] into the sum of the Xi, namely, as the sort of functions f : [n] > Ui Xi that obey the condition f(i) C i^Xi.
Xi Xi = X1 x ... x Xn = { f : [n] > Ui Xi | f(i) C Xi for all i}.
Viewing these functions as relations f c JxJxX, where X = Ui Xi,
???
Another way to view these elements is as triples i^j^x such that i = j ???
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===6.35. Reducibility of Sign Relations===
===6.35. Reducibility of Sign Relations===
