Changes

'''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.

'''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 <math>n\!</math>-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>k\!</math>-place relations it is convenient to use the terms ''domain'' and ''quorum'' as references to the wider and narrower sets, respectively.

<pre>

For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> 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".

# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.

# The notation <math>{}^{\backprime\backprime} \operatorname{Quo}_j (L) {}^{\prime\prime}\!</math> denotes a subset of <math>X_j\!</math> called the ''quorum of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> quorum of <math>L,\!</math>'' defined as follows.

2. The notation "Quoi (R)" denotes a subset of Xi called the "quorum of R at i" or the "ith quorum of R", defined as follows:

{| align="center" cellspacing="8" width="90%"

 

| <math>\operatorname{Quo}_j (L)\!</math>

Quoi (R) = the largest Q c Xi such that R&Q@i is >1-regular at i,

| =

= the largest Q c Xi such that |R&x@i| > 1 for all x C Q c Xi.

| the largest <math>Q \subseteq X_j\!</math> such that <math>L_{Q \,\text{at}\, j}\!</math> is <math>(> 1)\text{-regular at}~ j,\!</math>

| =

| the largest <math>Q \subseteq X_j\!</math> such that <math>|L_{Q \,\text{at}\, j}| > 1\!</math> for all <math>x \in Q \subseteq X_j.\!</math>

In the special case of a dyadic relation R c X1xX2 = SxT, including the case of a partial function p : S ~> T or a total function f : S  > T, I will stick to the following conventions:

In the special case of a dyadic relation R c X1xX2 = SxT, including the case of a partial function p : S ~> T or a total function f : S  > T, I will stick to the following conventions: