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A few more comments on terminology are needed in further preparation.  One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and <math>n\!</math>-place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.

A few more comments on terminology are needed in further preparation.  One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and n place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.

Up to this point in the discussion, the analysis of the A and B dialogue has concerned itself almost exclusively with the relationship of triadic sign relations to the dyadic relations obtained from them by taking their projections onto various relational planes.  In particular, a major focus of interest was the extent to which salient properties of sign relations can be gleaned from a study of their dyadic projections.

Up to this point in the discussion, the analysis of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue has concerned itself almost exclusively with the relationship of triadic sign relations to the dyadic relations obtained from them by taking their projections onto various relational planes.  In particular, a major focus of interest was the extent to which salient properties of sign relations can be gleaned from a study of their dyadic projections.

Two important topics for later discussion will be concerned with:  (1) the sense in which every n place relation can be decomposed in terms of triadic relations, and (2) the fact that not every triadic relation can be further reduced to conjunctions of dyadic relations.

Two important topics for later discussion will be concerned with:  (1) the sense in which every <math>n\!</math>-place relation can be decomposed in terms of triadic relations, and (2) the fact that not every triadic relation can be further reduced to conjunctions of dyadic relations.

It is one of the constant technical needs of this project to maintain a flexible language for talking about relations, one that permits discussion to shift from functional to relational emphases and from dyadic relations to n place relations with a maximum of ease.  It is not possible to do this without violating the favored conventions of one technical linguistic community or another.  I have chosen a strategy of use that respects as many different usages as possible, but in the end it cannot help but to reflect a few personal choices.  To some extent my choices are guided by an interest in developing the information, computation, and decision theoretic aspects of the mathematical language used.  Eventually, this requires one to render every distinction, even that of appearing or not in a particular category, as being relative to an interpretive framework.

'''Variant.'''  It is one of the constant technical needs of this project to maintain a flexible language for talking about relations, one that permits discussion to shift from functional to relational emphases and from dyadic relations to <math>n\!</math>-place relations with a maximum of ease.  It is not possible to do this without violating the favored conventions of one technical linguistic community or another.  I have chosen a strategy of use that respects as many different usages as possible, but in the end it cannot help but to reflect a few personal choices.  To some extent my choices are guided by an interest in developing the information, computation, and decision-theoretic aspects of the mathematical language used.  Eventually, this requires one to render every distinction, even that of appearing or not in a particular category, as being relative to an interpretive framework.

While operating in this context, it is necessary to distinguish "domains" in the broad sense from "domains of definition" in the narrow sense.  For n place relations it is convenient to use the terms "domain" and "quorum" as references to the wider and narrower sets, respectively.

While operating in this context, it is necessary to distinguish ''domains'' in the broad sense from ''domains of definition'' in the narrow sense.  For <math>n\!</math>-place relations it is convenient to use the terms ''domain'' and ''quorum'' as references to the wider and narrower sets, respectively.

For an n place relation R c X1x...xXn, I maintain the following usages:

For an n-place relation R c X1x...xXn, I maintain the following usages:

1. The notation "Domi (R)" denotes the set Xi, called the "domain of R at i" or the "ith domain of R".

1. The notation "Domi (R)" denotes the set Xi, called the "domain of R at i" or the "ith domain of R".