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→‎6.27. Differential Logic and Group Operations: format tables
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Using the appropriate isomorphisms, or recognizing how, in terms of the information given, that each of several descriptions is tantamount to the same object, the triadic relation <math>\ominus \subseteq X \times \operatorname{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \operatorname{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \operatorname{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\operatorname{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>
 
Using the appropriate isomorphisms, or recognizing how, in terms of the information given, that each of several descriptions is tantamount to the same object, the triadic relation <math>\ominus \subseteq X \times \operatorname{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \operatorname{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \operatorname{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\operatorname{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>
   −
Question.  Group Actions?
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<br>
   −
{| align="center" cellspacing="8" width="90%"
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'''Question.'''  Group Actions?  <math>r : \operatorname{d}X \to (X \to X)\!</math>
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| <math>r : \operatorname{d}X \to (X \to X).\!</math>
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 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" |
 +
<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{A} (V_4)\!</math>
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|- style="background:#f0f0ff"
 +
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
 +
| width="36%" | <math>\begin{matrix} \text{Logical}  \\ \text{Element} \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{List}    \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{Term}    \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Genetic}  \\ \text{Element} \end{matrix}</math>
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|-
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
r
 +
\\[4pt]
 +
s
 +
\\[4pt]
 +
t
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
(\operatorname{d}\underline{\underline{\text{a}}})
 +
(\operatorname{d}\underline{\underline{\text{b}}})
 +
(\operatorname{d}\underline{\underline{\text{i}}})
 +
(\operatorname{d}\underline{\underline{\text{u}}})
 +
\\[4pt]
 +
~\operatorname{d}\underline{\underline{\text{a}}}~
 +
(\operatorname{d}\underline{\underline{\text{b}}})
 +
~\operatorname{d}\underline{\underline{\text{i}}}~
 +
(\operatorname{d}\underline{\underline{\text{u}}})
 +
\\[4pt]
 +
(\operatorname{d}\underline{\underline{\text{a}}})
 +
~\operatorname{d}\underline{\underline{\text{b}}}~
 +
(\operatorname{d}\underline{\underline{\text{i}}})
 +
~\operatorname{d}\underline{\underline{\text{u}}}~
 +
\\[4pt]
 +
~\operatorname{d}\underline{\underline{\text{a}}}~
 +
~\operatorname{d}\underline{\underline{\text{b}}}~
 +
~\operatorname{d}\underline{\underline{\text{i}}}~
 +
~\operatorname{d}\underline{\underline{\text{u}}}~
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\langle \operatorname{d}! \rangle
 +
\\[4pt]
 +
\langle
 +
\operatorname{d}\underline{\underline{\text{a}}} ~
 +
\operatorname{d}\underline{\underline{\text{i}}}
 +
\rangle
 +
\\[4pt]
 +
\langle
 +
\operatorname{d}\underline{\underline{\text{b}}} ~
 +
\operatorname{d}\underline{\underline{\text{u}}}
 +
\rangle
 +
\\[4pt]
 +
\langle \operatorname{d}* \rangle
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\operatorname{d}!
 +
\\[4pt]
 +
\operatorname{d}\underline{\underline{\text{a}}} \cdot
 +
\operatorname{d}\underline{\underline{\text{i}}} ~ !
 +
\\[4pt]
 +
\operatorname{d}\underline{\underline{\text{b}}} \cdot
 +
\operatorname{d}\underline{\underline{\text{u}}} ~ !
 +
\\[4pt]
 +
\operatorname{d}*
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\end{matrix}</math>
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| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
\operatorname{d}_{\text{ai}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{bu}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{ai}} * \operatorname{d}_{\text{bu}}
 +
\end{matrix}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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|+ style="height:30px" |
 +
<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{B} (V_4)\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
 +
| width="36%" | <math>\begin{matrix} \text{Logical}  \\ \text{Element} \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{List}    \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{Term}    \end{matrix}</math>
 +
| width="16%" | <math>\begin{matrix} \text{Genetic}  \\ \text{Element} \end{matrix}</math>
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|-
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
r
 +
\\[4pt]
 +
s
 +
\\[4pt]
 +
t
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
(\operatorname{d}\underline{\underline{\text{a}}})
 +
(\operatorname{d}\underline{\underline{\text{b}}})
 +
(\operatorname{d}\underline{\underline{\text{i}}})
 +
(\operatorname{d}\underline{\underline{\text{u}}})
 +
\\[4pt]
 +
~\operatorname{d}\underline{\underline{\text{a}}}~
 +
(\operatorname{d}\underline{\underline{\text{b}}})
 +
(\operatorname{d}\underline{\underline{\text{i}}})
 +
~\operatorname{d}\underline{\underline{\text{u}}}~
 +
\\[4pt]
 +
(\operatorname{d}\underline{\underline{\text{a}}})
 +
~\operatorname{d}\underline{\underline{\text{b}}}~
 +
~\operatorname{d}\underline{\underline{\text{i}}}~
 +
(\operatorname{d}\underline{\underline{\text{u}}})
 +
\\[4pt]
 +
~\operatorname{d}\underline{\underline{\text{a}}}~
 +
~\operatorname{d}\underline{\underline{\text{b}}}~
 +
~\operatorname{d}\underline{\underline{\text{i}}}~
 +
~\operatorname{d}\underline{\underline{\text{u}}}~
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\langle \operatorname{d}! \rangle
 +
\\[4pt]
 +
\langle
 +
\operatorname{d}\underline{\underline{\text{a}}} ~
 +
\operatorname{d}\underline{\underline{\text{u}}}
 +
\rangle
 +
\\[4pt]
 +
\langle
 +
\operatorname{d}\underline{\underline{\text{b}}} ~
 +
\operatorname{d}\underline{\underline{\text{i}}}
 +
\rangle
 +
\\[4pt]
 +
\langle \operatorname{d}* \rangle
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\operatorname{d}!
 +
\\[4pt]
 +
\operatorname{d}\underline{\underline{\text{a}}} \cdot
 +
\operatorname{d}\underline{\underline{\text{u}}} ~ !
 +
\\[4pt]
 +
\operatorname{d}\underline{\underline{\text{b}}} \cdot
 +
\operatorname{d}\underline{\underline{\text{i}}} ~ !
 +
\\[4pt]
 +
\operatorname{d}*
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
\operatorname{d}_{\text{au}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{bi}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{au}} * \operatorname{d}_{\text{bi}}
 +
\end{matrix}</math>
 
|}
 
|}
   −
<pre>
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<br>
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Table 42.1  Group Representation RepA (V4)
  −
Abstract Logical Active Active Genetic
  −
Element Element List Term Element
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1 (da)(db)(di)(du) <d!> d! 1
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r da (db) di (du) <da di> da.di! dai
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s (da) db (di) du <db du> db.du! dbu
  −
t da  db  di  du <d*> d* dai*dbu
  −
</pre>
     −
<pre>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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Table 42.2  Group Representation RepB (V4)
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|+ style="height:30px" |
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Abstract Logical Active Active Genetic
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<math>\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)\!</math>
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Element Element List Term Element
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|- style="background:#f0f0ff"
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1 (da)(db)(di)(du) <d!> d! 1
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| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
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r da (db)(di) du <da du> da.du! dau
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| width="36%" | <math>\begin{matrix} \text{Logical}  \\ \text{Element} \end{matrix}</math>
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s (da) db  di (du) <db di> db.di! dbi
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| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{List}    \end{matrix}</math>
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t da  db  di  du <d*> d* dau*dbi
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| width="16%" | <math>\begin{matrix} \text{Active}  \\ \text{Term}    \end{matrix}</math>
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</pre>
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| width="16%" | <math>\begin{matrix} \text{Genetic}  \\ \text{Element} \end{matrix}</math>
 +
|-
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
r
 +
\\[4pt]
 +
s
 +
\\[4pt]
 +
t
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
(\operatorname{d}\text{m})
 +
(\operatorname{d}\text{n})
 +
\\[4pt]
 +
~\operatorname{d}\text{m}~
 +
(\operatorname{d}\text{n})
 +
\\[4pt]
 +
(\operatorname{d}\text{m})
 +
~\operatorname{d}\text{n}~
 +
\\[4pt]
 +
~\operatorname{d}\text{m}~
 +
~\operatorname{d}\text{n}~
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\langle\operatorname{d}!\rangle
 +
\\[4pt]
 +
\langle\operatorname{d}\text{m}\rangle
 +
\\[4pt]
 +
\langle\operatorname{d}\text{n}\rangle
 +
\\[4pt]
 +
\langle\operatorname{d}*\rangle
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\operatorname{d}!
 +
\\[4pt]
 +
\operatorname{d}\text{m}!
 +
\\[4pt]
 +
\operatorname{d}\text{n}!
 +
\\[4pt]
 +
\operatorname{d}*
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
1
 +
\\[4pt]
 +
\operatorname{d}_{\text{m}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{n}}
 +
\\[4pt]
 +
\operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}}
 +
\end{matrix}</math>
 +
|}
   −
<pre>
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<br>
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Table 42.3  Group Representation RepC (V4)
  −
Abstract Logical Active Active Genetic
  −
Element Element List Term Element
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1 (dm)(dn) <d!> d! 1
  −
r dm (dn) <dm> dm! dm
  −
s (dm) dn <dn> dn! dn
  −
t dm  dn <d*> d* dm*dn
  −
</pre>
      
<pre>
 
<pre>
Jon Awbrey
12,080

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