Changes

One way to approach the formalization of an objective genre ''G'' is through an indexed collection of dyadic relations:

One way to approach the formalization of an objective genre ''G'' is through an indexed collection of dyadic relations:

:<p><math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq P_j \times Q_j\ \mbox{for all}\ j \in J .</math></p>

:<p><math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq P_j \times Q_j\ (\forall j \in J) .</math></p>

Here, ''J'' is a set of actual (not formal) parameters used to index the OG, while ''P''_''j'' and ''Q''_''j'' are domains of objects (initially in the informal sense) that enter into the dyadic relations ''G''_''j''&nbsp;.

Here, ''J'' is a set of actual (not formal) parameters used to index the OG, while ''P''_''j'' and ''Q''_''j'' are domains of objects (initially in the informal sense) that enter into the dyadic relations ''G''_''j''&nbsp;.

: Rubric of Universal Inclusion (RUI):  <math>X = \textstyle \bigcup_j (P_j \cup Q_j) .</math>

: Rubric of Universal Inclusion (RUI):  <math>X = \textstyle \bigcup_j (P_j \cup Q_j) .</math>

: Rubric of Universal Equality (RUE):  <math>X = P_j = Q_j\ \mbox{for all}\ j \in J .</math>

: Rubric of Universal Equality (RUE):  <math>X = P_j = Q_j\ (\forall j \in J) .</math>

Working under either of these assumptions, ''G'' can be provided with a simplified form of presentation:

Working under either of these assumptions, ''G'' can be provided with a simplified form of presentation:

:<p><math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq X \times X\ \mbox{for all}\ j \in J .</math></p>

:<p><math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq X \times X\ (\forall j \in J) .</math></p>

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation.  Generally speaking, it is always possible in principle to form the union required by the RUI, or without loss of generality to assume the equality imposed by the RUE.  The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context.  Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation.  Generally speaking, it is always possible in principle to form the union required by the RUI, or without loss of generality to assume the equality imposed by the RUE.  The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context.  Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

Often one's level of interest in a genre is ''purely generic''.  When the relevant genre is regarded as an indexed family of dyadic relations, ''G'' = {''G''_''j''}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

Often one's level of interest in a genre is ''purely generic''.  When the relevant genre is regarded as an indexed family of dyadic relations, ''G'' = {''G''_''j''}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

:<p><math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j\ \mbox{for some}\ j \in J \} .</math></p>

:<p><math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j\ (\exists j \in J) \} .</math></p>

When the relevant genre is contemplated as a triadic relation, ''G''&nbsp;&sube;&nbsp;''J''&nbsp;&times;&nbsp;''X''&nbsp;&times;&nbsp;''X'', then one is dealing with the projection of ''G'' on the object dyad ''X''&nbsp;&times;&nbsp;''X''.

When the relevant genre is contemplated as a triadic relation, ''G''&nbsp;&sube;&nbsp;''J''&nbsp;&times;&nbsp;''X''&nbsp;&times;&nbsp;''X'', then one is dealing with the projection of ''G'' on the object dyad ''X''&nbsp;&times;&nbsp;''X''.

: GXX  = ProjXX(G) = {‹x, y› ? X?X : ‹j, x, y› ? G for some j ? J}.

:<p><math>G_{XX} = proj_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G\ (\exists j \in J) \} .</math></p>

On these occasions, the assertion that ‹x, y›  ?  UJG  = GXX  can be indicated by any one of the following equivalent expressions:

On these occasions, the assertion that (''x'',&nbsp;''y'') &isin; <font size="+2">&cup;</font>_''J''&nbsp;''G'' = ''G''_''XX'' can be indicated by any one of the following equivalent expressions:

: G : x < y, x <G y, x < y : G,

: G : x < y, x <G y, x < y : G,