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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 22:24, 1 December 2015

21,335 bytes added ,  22:24, 1 December 2015
→‎Operational Representation: + a few more paragraphs
Line 2,964: Line 2,964:     
<br>
 
<br>
 +
 +
A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.\!</math>
 +
 +
In the case of <math>V_4 = (G, \cdot),\!</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h} \},\!</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G\!</math> whose triples are listed below:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
(\mathrm{e}, \mathrm{e}, \mathrm{e}) &
 +
(\mathrm{e}, \mathrm{f}, \mathrm{f}) &
 +
(\mathrm{e}, \mathrm{g}, \mathrm{g}) &
 +
(\mathrm{e}, \mathrm{h}, \mathrm{h})
 +
\\[6pt]
 +
(\mathrm{f}, \mathrm{e}, \mathrm{f}) &
 +
(\mathrm{f}, \mathrm{f}, \mathrm{e}) &
 +
(\mathrm{f}, \mathrm{g}, \mathrm{h}) &
 +
(\mathrm{f}, \mathrm{h}, \mathrm{g})
 +
\\[6pt]
 +
(\mathrm{g}, \mathrm{e}, \mathrm{g}) &
 +
(\mathrm{g}, \mathrm{f}, \mathrm{h}) &
 +
(\mathrm{g}, \mathrm{g}, \mathrm{e}) &
 +
(\mathrm{g}, \mathrm{h}, \mathrm{f})
 +
\\[6pt]
 +
(\mathrm{h}, \mathrm{e}, \mathrm{h}) &
 +
(\mathrm{h}, \mathrm{f}, \mathrm{g}) &
 +
(\mathrm{h}, \mathrm{g}, \mathrm{f}) &
 +
(\mathrm{h}, \mathrm{h}, \mathrm{e})
 +
\end{matrix}\!</math>
 +
|}
 +
 +
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3\!</math> is actually a function <math>L : G \times G \to G.\!</math>  It is from this functional perspective that we can see an easy way to derive the two regular representations.  Since we have a function of the type <math>L : G \times G \to G,\!</math> we can define a couple of substitution operators:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| <math>\mathrm{Sub}(x, (\underline{~~}, y))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),\!</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.~\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\mathrm{Sub}(x, (y, \underline{~~}))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),\!</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.~\!</math>
 +
|}
 +
 +
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)\!</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.\!</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin.  This aspect of pragmatic definition we recognize as the regular ante-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
\end{matrix}\!</math>
 +
|}
 +
 +
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})\!</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.\!</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin.  This aspect of pragmatic definition we recognize as the regular post-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
\end{matrix}\!</math>
 +
|}
 +
 +
If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.
 +
 +
A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.\!</math>
 +
 +
In the case of <math>V_4 = (G, \cdot),\!</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h} \},\!</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G\!</math> whose triples are listed below:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
(\mathrm{e}, \mathrm{e}, \mathrm{e}) &
 +
(\mathrm{e}, \mathrm{f}, \mathrm{f}) &
 +
(\mathrm{e}, \mathrm{g}, \mathrm{g}) &
 +
(\mathrm{e}, \mathrm{h}, \mathrm{h})
 +
\\[6pt]
 +
(\mathrm{f}, \mathrm{e}, \mathrm{f}) &
 +
(\mathrm{f}, \mathrm{f}, \mathrm{e}) &
 +
(\mathrm{f}, \mathrm{g}, \mathrm{h}) &
 +
(\mathrm{f}, \mathrm{h}, \mathrm{g})
 +
\\[6pt]
 +
(\mathrm{g}, \mathrm{e}, \mathrm{g}) &
 +
(\mathrm{g}, \mathrm{f}, \mathrm{h}) &
 +
(\mathrm{g}, \mathrm{g}, \mathrm{e}) &
 +
(\mathrm{g}, \mathrm{h}, \mathrm{f})
 +
\\[6pt]
 +
(\mathrm{h}, \mathrm{e}, \mathrm{h}) &
 +
(\mathrm{h}, \mathrm{f}, \mathrm{g}) &
 +
(\mathrm{h}, \mathrm{g}, \mathrm{f}) &
 +
(\mathrm{h}, \mathrm{h}, \mathrm{e})
 +
\end{matrix}\!</math>
 +
|}
 +
 +
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3\!</math> is actually a function <math>L : G \times G \to G.\!</math>  It is from this functional perspective that we can see an easy way to derive the two regular representations.  Since we have a function of the type <math>L : G \times G \to G,\!</math> we can define a couple of substitution operators:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| <math>\mathrm{Sub}(x, (\underline{~~}, y))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),\!</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.~\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\mathrm{Sub}(x, (y, \underline{~~}))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),\!</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.~\!</math>
 +
|}
 +
 +
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)\!</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.\!</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin.  This aspect of pragmatic definition we recognize as the regular ante-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
\end{matrix}\!</math>
 +
|}
 +
 +
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})\!</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.\!</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin.  This aspect of pragmatic definition we recognize as the regular post-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
\end{matrix}\!</math>
 +
|}
 +
 +
If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.
 +
 +
So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, <math>G = \{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ a, b, c \},\!</math> usually notated as <math>G = \mathrm{Sym}(X)\!</math> or more abstractly and briefly, as <math>\mathrm{Sym}(3)\!</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\mathrm{Sym}(X).\!</math>
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ a, b, c \}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\mathrm{e}\!</math>
 +
| width="16%" | <math>\mathrm{f}\!</math>
 +
| width="16%" | <math>\mathrm{g}\!</math>
 +
| width="16%" | <math>\mathrm{h}\!</math>
 +
| width="16%" | <math>\mathrm{i}~\!</math>
 +
| width="16%" | <math>\mathrm{j}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & b & c
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & a & b
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & c & a
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & c & b
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & b & a
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & a & c
 +
\end{matrix}\!</math>
 +
|}
 +
 +
<br>
 +
 +
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
 +
 +
{| align="center" cellpadding="10" style="text-align:center"
 +
| <math>\text{Symmetric Group}~ S_3\!</math>
 +
|-
 +
| [[Image:Symmetric Group S(3).jpg|500px]]
 +
|}
 +
 +
By the way, we will meet with the symmetric group <math>S_3~\!</math> again when we return to take up the study of Peirce's early paper &ldquo;On a Class of Multiple Algebras&rdquo; (CP&nbsp;3.324&ndash;327), and also his late unpublished work &ldquo;The Simplest Mathematics&rdquo; (1902) (CP&nbsp;4.227&ndash;323), with particular reference to the section that treats of &ldquo;Trichotomic Mathematics&rdquo; (CP&nbsp;4.307&ndash;323).
 +
 +
By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\mathrm{Sym}(3).\!</math>  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early &ldquo;Algebra + Logic&rdquo; papers.
 +
 +
Writing the permutations or substitutions of <math>\mathrm{Sym} \{ a, b, c \}\!</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.~\!</math>
 +
 +
{| align="center" cellpadding="10" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\mathrm{e}
 +
& = & a\!:\!a
 +
& + & b\!:\!b
 +
& + & c\!:\!c
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & a\!:\!c
 +
& + & b\!:\!a
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & a\!:\!b
 +
& + & b\!:\!c
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & a\!:\!a
 +
& + & b\!:\!c
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\mathrm{i}
 +
& = & a\!:\!c
 +
& + & b\!:\!b
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\mathrm{j}
 +
& = & a\!:\!b
 +
& + & b\!:\!a
 +
& + & c\!:\!c
 +
\end{matrix}\!</math>
 +
|}
 +
 +
I have without stopping to think about it written out this natural representation of <math>S_3~\!</math> in the style that comes most naturally to me, to wit, the &ldquo;right&rdquo; way, whereby an ordered pair configured as <math>x\!:\!y\!</math> constitutes the turning of <math>x\!</math> into <math>y.\!</math>  It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
 +
 +
To construct the regular representations of <math>S_3,~\!</math> we begin with the data of its operation table:
 +
 +
{| align="center" cellpadding="10" style="text-align:center"
 +
| <math>\text{Symmetric Group}~ S_3\!</math>
 +
|-
 +
| [[Image:Symmetric Group S(3).jpg|500px]]
 +
|}
 +
 +
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
 +
 +
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3\!</math> is actually a function <math>L : G \times G \to G.\!</math>  It is from this functional perspective that we can see an easy way to derive the two regular representations.
 +
 +
Since we have a function of the type <math>L : G \times G \to G,\!</math> we can define a couple of substitution operators:
 +
 +
{| align="center" cellpadding="10" width="90%"
 +
| valign="top" | 1.
 +
| <math>\mathrm{Sub}(x, (\underline{~~}, y))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),\!</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.~\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\mathrm{Sub}(x, (y, \underline{~~}))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),\!</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.~\!</math>
 +
|}
 +
 +
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)\!</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.\!</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin.  This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
 +
 +
{| align="center" cellpadding="10" style="text-align:center"
 +
|
 +
<math>\begin{array}{*{13}{c}}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
& + & \mathrm{i}\!:\!\mathrm{i}
 +
& + & \mathrm{j}\!:\!\mathrm{j}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{j}
 +
& + & \mathrm{i}\!:\!\mathrm{h}
 +
& + & \mathrm{j}\!:\!\mathrm{i}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{i}
 +
& + & \mathrm{i}\!:\!\mathrm{j}
 +
& + & \mathrm{j}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{i}
 +
& + & \mathrm{g}\!:\!\mathrm{j}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
& + & \mathrm{i}\!:\!\mathrm{f}
 +
& + & \mathrm{j}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{i}
 +
& = & \mathrm{e}\!:\!\mathrm{i}
 +
& + & \mathrm{f}\!:\!\mathrm{j}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
& + & \mathrm{i}\!:\!\mathrm{e}
 +
& + & \mathrm{j}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{j}
 +
& = & \mathrm{e}\!:\!\mathrm{j}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{i}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
& + & \mathrm{i}\!:\!\mathrm{g}
 +
& + & \mathrm{j}\!:\!\mathrm{e}
 +
\end{array}\!</math>
 +
|}
 +
 +
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})\!</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.\!</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin.  This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
 +
 +
{| align="center" cellpadding="10" style="text-align:center"
 +
|
 +
<math>\begin{array}{*{13}{c}}
 +
\mathrm{e}
 +
& = & \mathrm{e}\!:\!\mathrm{e}
 +
& + & \mathrm{f}\!:\!\mathrm{f}
 +
& + & \mathrm{g}\!:\!\mathrm{g}
 +
& + & \mathrm{h}\!:\!\mathrm{h}
 +
& + & \mathrm{i}\!:\!\mathrm{i}
 +
& + & \mathrm{j}\!:\!\mathrm{j}
 +
\\[4pt]
 +
\mathrm{f}
 +
& = & \mathrm{e}\!:\!\mathrm{f}
 +
& + & \mathrm{f}\!:\!\mathrm{g}
 +
& + & \mathrm{g}\!:\!\mathrm{e}
 +
& + & \mathrm{h}\!:\!\mathrm{i}
 +
& + & \mathrm{i}\!:\!\mathrm{j}
 +
& + & \mathrm{j}\!:\!\mathrm{h}
 +
\\[4pt]
 +
\mathrm{g}
 +
& = & \mathrm{e}\!:\!\mathrm{g}
 +
& + & \mathrm{f}\!:\!\mathrm{e}
 +
& + & \mathrm{g}\!:\!\mathrm{f}
 +
& + & \mathrm{h}\!:\!\mathrm{j}
 +
& + & \mathrm{i}\!:\!\mathrm{h}
 +
& + & \mathrm{j}\!:\!\mathrm{i}
 +
\\[4pt]
 +
\mathrm{h}
 +
& = & \mathrm{e}\!:\!\mathrm{h}
 +
& + & \mathrm{f}\!:\!\mathrm{j}
 +
& + & \mathrm{g}\!:\!\mathrm{i}
 +
& + & \mathrm{h}\!:\!\mathrm{e}
 +
& + & \mathrm{i}\!:\!\mathrm{g}
 +
& + & \mathrm{j}\!:\!\mathrm{f}
 +
\\[4pt]
 +
\mathrm{i}
 +
& = & \mathrm{e}\!:\!\mathrm{i}
 +
& + & \mathrm{f}\!:\!\mathrm{h}
 +
& + & \mathrm{g}\!:\!\mathrm{j}
 +
& + & \mathrm{h}\!:\!\mathrm{f}
 +
& + & \mathrm{i}\!:\!\mathrm{e}
 +
& + & \mathrm{j}\!:\!\mathrm{g}
 +
\\[4pt]
 +
\mathrm{j}
 +
& = & \mathrm{e}\!:\!\mathrm{j}
 +
& + & \mathrm{f}\!:\!\mathrm{i}
 +
& + & \mathrm{g}\!:\!\mathrm{h}
 +
& + & \mathrm{h}\!:\!\mathrm{g}
 +
& + & \mathrm{i}\!:\!\mathrm{f}
 +
& + & \mathrm{j}\!:\!\mathrm{e}
 +
\end{array}\!</math>
 +
|}
 +
 +
If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3~\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.
 +
 +
{| cellpadding="2" cellspacing="2" width="100%"
 +
| width="60%" | &nbsp;
 +
| width="40%" |
 +
the way of heaven and earth<br>
 +
is to be long continued<br>
 +
in their operation<br>
 +
without stopping
 +
|-
 +
| height="50px" | &nbsp;
 +
| valign="top"  | &mdash; i ching, hexagram 32
 +
|}
    
==Logical Cacti==
 
==Logical Cacti==
Jon Awbrey
12,080

edits

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