Changes

For some reason the ultimately obvious method seldom presents itself exactly in this wise without diligent work on the part of the inquirer, or one who would arrogate the roles of both its former and its follower.  Perhaps this has to do with the problematic role of ''synthetic a priori'' truths in constructive mathematics.  Perhaps the mystery lies encrypted, no doubt buried in some obscure dead letter office, due to the obliterate indicia on the letters "P", "Q", and "X" inscribed above.  No matter - at the moment there are far more pressing rounds to make.

For some reason the ultimately obvious method seldom presents itself exactly in this wise without diligent work on the part of the inquirer, or one who would arrogate the roles of both its former and its follower.  Perhaps this has to do with the problematic role of ''synthetic a priori'' truths in constructive mathematics.  Perhaps the mystery lies encrypted, no doubt buried in some obscure dead letter office, due to the obliterate indicia on the letters "P", "Q", and "X" inscribed above.  No matter - at the moment there are far more pressing rounds to make.

Given a genre G whose OM's are indexed by a set J and whose objects form a set X, there is a triadic relation among an OM and a pair of objects that exists when the first object belongs to the second object according to that OM.  This is called the "standing relation" of the OG, and it can be taken as one way of defining and establishing the genre.  In the way that triadic relations usually give rise to dyadic operations, the associated ''standing operation'' of the OG can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated OM.

 

Given a genre ''G'' whose OM's are indexed by a set ''J'' and whose objects form a set ''X'', there is a triadic relation among an OM and a pair of objects that exists when the first object belongs to the second object according to that OM.  This is called the ''standing relation'' of the OG, and it can be taken as one way of defining and establishing the genre.  In the way that triadic relations usually give rise to dyadic operations, the associated ''standing operation'' of the OG can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated OM.

 

There is a ''partial converse'' of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the ''propping relation'' of the OG, and it can be taken as an alternate way of defining the genre.

There is a ''partial converse'' of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the ''propping relation'' of the OG, and it can be taken as an alternate way of defining the genre.

G^  = {‹j, q, p› ? J?Q?P : ‹j, p, q› ? G}, or

:<p><math>G\!\uparrow \ = \ \{(j, q, p) \in J \times Q \times P : (j, p, q) \in G \} ,</math></p>

 

: or

G^  = {‹j, y, x› ? J?X?X : ‹j, x, y› ? G}.

:<p><math>G\!\uparrow \ = \ \{(j, y, x) \in J \times X \times X : (j, x, y) \in G \} .</math></p>

The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an OG:

The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an OG:

1. The standing relation of an OG is denoted by the symbol ":<", pronounced ''set-in'', so that  :< ? JxPxQ  or  :< ? J?X?X.

# The standing relation of an OG is denoted by the symbol ":<", pronounced ''set-in'', so that  :< ? JxPxQ  or  :< ? J?X?X.

 

# The propping relation of an OG is denoted by the symbol ":>", pronounced ''set-on'', so that  :> ? J?Q?P  or  :> ? J?X?X.

2. The propping relation of an OG is denoted by the symbol ":>", pronounced ''set-on'', so that  :> ? J?Q?P  or  :> ? J?X?X.

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 = {Gj}, 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 = {Gj}, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

UJG  =  Uj Gj  =  {‹x, y› ? X?X : ‹x, y› ? Gj for some j ? J}.

: UJG  =  Uj Gj  =  {‹x, y› ? X?X : ‹x, y› ? Gj for some j ? J}.

When the relevant genre is contemplated as a triadic relation, G ? J?X?X, then one is dealing with the projection of G on the object dyad XxX.

When the relevant genre is contemplated as a triadic relation, G ? J?X?X, then one is dealing with the projection of G on the object dyad XxX.

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

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

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, y›  ?  UJG  =  GXX  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,

G : y > x, y >G x, y > x : G.

: G : y > x, y >G x, y > x : G.

At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation (IDR) that links two objects.  To indicate that a triple consisting of an OM j and two objects x and y belongs to the standing relation of the OG, ‹j, x, y› ? :<, or equally, to indicate that a triple consisting of an OM j and two objects y and x belongs to the propping relation of the OG, ‹j, y, x› ? :>, all of the following notations are equivalent:

At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation (IDR) that links two objects.  To indicate that a triple consisting of an OM j and two objects x and y belongs to the standing relation of the OG, ‹j, x, y› ? :<, or equally, to indicate that a triple consisting of an OM j and two objects y and x belongs to the propping relation of the OG, ‹j, y, x› ? :>, all of the following notations are equivalent:

j : x < y, x <j y, x < y : j,

: j : x < y, x <j y, x < y : j,

j : y > x, y >j x, y > x : j.

: j : y > x, y >j x, y > x : j.

Assertions of these relations can be read in various ways, for example:

Assertions of these relations can be read in various ways, for example: