Changes

A '''<math>k\!</math>-ary relation''' or a '''<math>k\!</math>-place relation''' is a relation on an ordered <math>k\!</math>-tuple of nonempty sets.  Thus, <math>L\!</math> is a <math>k\!</math>-place relation relation on the SOD <math>(X_1, \ldots, X_k)\!</math> if and only if <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>  In various applications, the <math>k\!</math>-tuple elements <math>(x_1, \ldots, x_k)\!</math> of <math>L\!</math> are called its ''elementary relations'', ''individual transactions'', ''ingredients'', or ''effects''.

A '''<math>k\!</math>-ary relation''' or a '''<math>k\!</math>-place relation''' is a relation on an ordered <math>k\!</math>-tuple of nonempty sets.  Thus, <math>L\!</math> is a <math>k\!</math>-place relation relation on the SOD <math>(X_1, \ldots, X_k)\!</math> if and only if <math>L \subseteq X_1 \times \ldots \times X_k.\!</math>  In various applications, the <math>k\!</math>-tuple elements <math>(x_1, \ldots, x_k)\!</math> of <math>L\!</math> are called its ''elementary relations'', ''individual transactions'', ''ingredients'', or ''effects''.

Before continuing with the chain of definitions, a slight digression is needed at this point to loosen up the interpretation of relation symbols in what follows.  Exercising a certain amount of flexibility with notation, and relying on a discerning interpretation of equivocal expressions, one can use the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> or any other indication of a <math>k\!</math>-place relation <math>L\!</math> in a wide variety of different fashions, both logical and operational.

Before continuing with the chain of definitions, a slight digression is needed at this point to loosen up the interpretation of relation symbols in what follows.  Exercising a certain amount of flexibility with notation, and relying on a discerning interpretation of equivocal expressions, one can use the name "R" or any other indication of an n place relation R in a wide variety of different fashions, both logical and operational.

First, R can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others.  In this way, "R" can be interpreted as naming a function from Xi Xi to the domain of truth values B = {0, 1}.  With the appropriate understanding, it is permissible to write "R : X1x...xXn  > B" to indicate this interpretation.

First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others.  In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.

Second, R can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects.  In particular, if one is given a partial effect or an incomplete n tuple, say, one that is missing a value in the jth place, as indicated by the notation "<x1, ...,  j, ..., xn>", then "R" can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place.  With this in mind, it is permissible to write "R : X1x...x jx...xXn  > Pow(Xj)" to indicate this use of "R".  If the sets in the range of this function are all singletons, then it is permissible to write "R : X1x...x jx...xXn  > Xj" to specify the corresponding use of "R".

Second, R can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects.  In particular, if one is given a partial effect or an incomplete n tuple, say, one that is missing a value in the jth place, as indicated by the notation "<x1, ...,  j, ..., xn>", then "R" can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place.  With this in mind, it is permissible to write "R : X1x...x jx...xXn  > Pow(Xj)" to indicate this use of "R".  If the sets in the range of this function are all singletons, then it is permissible to write "R : X1x...x jx...xXn  > Xj" to specify the corresponding use of "R".