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5,324 bytes added ,  01:25, 2 July 2007
→‎Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
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<pre>
 
<pre>
−
-------------o
  −
|      | f, g : U -> B          | Proposition,    | B^n -> B                  |
  −
|      |                        |  special case  |                            |
  −
| f    | f : U -> [x] c X%      |  of a mapping,  | c (B^n, B^n -> B)          |
  −
|      |                        |  or component  |                            |
  −
| g    | g : U -> [y] c X%      |  of a mapping.  | = (B^n +-> B) = [B^n]      |
   
-------------o
 
-------------o
 
| W    | W :                    | Operator        |                            |
 
| W    | W :                    | Operator        |                            |
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-------------o
 
</pre>
 
</pre>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
 +
|+ '''Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators'''
 +
|- style="background:paleturquoise"
 +
! Item
 +
! Notation
 +
! Description
 +
! Type
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup>
 +
| = [''u'', ''v'']
 +
| Source Universe
 +
| ['''B'''<sup>2</sup>]
 +
|-
 +
| ''X''<sup>&nbsp;&bull;</sup>
 +
| = [''x'']
 +
| Target Universe
 +
| ['''B'''<sup>1</sup>]
 +
|-
 +
| E''U''<sup>&nbsp;&bull;</sup>
 +
| = [''u'', ''v'', d''u'', d''v'']
 +
| Extended Source Universe
 +
| ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]
 +
|-
 +
| E''X''<sup>&nbsp;&bull;</sup>
 +
| = [''x'', d''x'']
 +
| Extended Target Universe
 +
| ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]
 +
|-
 +
| ''J''
 +
| ''J'' : ''U'' &rarr; '''B'''
 +
| Proposition
 +
| ('''B'''<sup>2</sup> &rarr; '''B''') &isin; ['''B'''<sup>2</sup>]
 +
|-
 +
| ''J''
 +
| ''J'' : ''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>
 +
| Transformation, or Mapping
 +
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>]
 +
|-
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| W
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| W&nbsp;:
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
 +
|-
 +
| &rarr;
 +
|-
 +
| (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,
 +
|-
 +
| for each W in the set:
 +
|-
 +
| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Operator
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
 +
| &nbsp;
 +
|-
 +
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]&nbsp;,
 +
|-
 +
| ['''B'''<sup>1</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]&nbsp;,
 +
|-
 +
| (['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>])
 +
|-
 +
| &rarr;
 +
|-
 +
| (['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>])
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <math>\epsilon</math>
 +
|-
 +
| <math>\eta</math>
 +
|-
 +
| E
 +
|-
 +
| D
 +
|-
 +
| d
 +
|}
 +
| valign="top" | &nbsp;
 +
| colspan="2"  |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
 +
| Tacit Extension Operator || <math>\epsilon</math>
 +
|-
 +
| Trope Extension Operator || <math>\eta</math>
 +
|-
 +
| Enlargement Operator    || E
 +
|-
 +
| Difference Operator      || D
 +
|-
 +
| Differential Operator    || d
 +
|}
 +
|-
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''W'''</font>
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''W'''</font>&nbsp;:
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
 +
|-
 +
| &rarr;
 +
|-
 +
| (<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>)&nbsp;,
 +
|-
 +
| for each <font face=georgia>'''W'''</font> in the set:
 +
|-
 +
| {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''T'''</font>}
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Operator
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
 +
| &nbsp;
 +
|-
 +
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]&nbsp;,
 +
|-
 +
| ['''B'''<sup>1</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]&nbsp;,
 +
|-
 +
| (['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>])
 +
|-
 +
| &rarr;
 +
|-
 +
| (['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>])
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''e'''</font>
 +
|-
 +
| <font face=georgia>'''E'''</font>
 +
|-
 +
| <font face=georgia>'''D'''</font>
 +
|-
 +
| <font face=georgia>'''T'''</font>
 +
|}
 +
| valign="top" | &nbsp;
 +
| colspan="2"  |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
 +
| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›
 +
|-
 +
| Secant Operator || <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›
 +
|-
 +
| Chord Operator  || <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›
 +
|-
 +
| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›
 +
|}
 +
|}<br>
    
===Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes===
 
===Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes===
Jon Awbrey
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