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 an objective genre <math>G\!</math> whose motives are indexed by a set <math>J\!</math> and whose objects form a set <math>X\!</math>, there is a triadic relation among a motive and a pair of objects that exists when the first object belongs to the second object according to that motive.  This is called the ''standing relation'' of the genre, 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 genre 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 motive.

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 genre, and it can be taken as an alternate way of defining the genre.

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The propping relation of an OG is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:

The propping relation of a genre is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:

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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, <math>G = \{ G_j \}</math>, 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, <math>G = \{ G_j \}\!</math>, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

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

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

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| ''j'' : ''x'' <math>\lessdot</math> ''y''

| <math>j : x \lessdot y</math>

| ''j'' : ''y'' <math>\gtrdot</math> ''x''

| <math>j : y \gtrdot x</math>

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|- style="background:ghostwhite"

| ''x'' <math>\lessdot</math>_''j'' ''y''

| <math>x \lessdot_j y</math>

| ''y'' <math>\gtrdot</math>_''j'' ''x''

| <math>y \gtrdot_j x</math>

|- style="background:paleturquoise"

|- style="background:ghostwhite"

| ''x'' <math>\lessdot</math> ''y'' : ''j''

| <math>x \lessdot y : j</math>

| ''y'' <math>\gtrdot</math> ''x'' : ''j''

| <math>y \gtrdot x : j</math>

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| ''j'' sets ''x'' in ''y''.

| ''j'' sets ''x'' in ''y''.