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Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 18:20, 4 June 2009
, 18:20, 4 June 2009→Note 21: markup
==Note 21==
==Note 21==
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<pre>
<pre>
Table 1. Symmetric Group S_3
Table 1. Symmetric Group S_3
o-------------------------------------------------o
| |
| ^ |
| e / \ e |
| / \ |
| / e \ |
| f / \ / \ f |
| / \ / \ |
| / f \ f \ |
| g / \ / \ / \ g |
| / \ / \ / \ |
| / g \ g \ g \ |
| h / \ / \ / \ / \ h |
| / \ / \ / \ / \ |
| / h \ e \ e \ h \ |
| i / \ / \ / \ / \ / \ i |
| / \ / \ / \ / \ / \ |
| / i \ i \ f \ j \ i \ |
| j / \ / \ / \ / \ / \ / \ j |
| / \ / \ / \ / \ / \ / \ |
| ( j \ j \ j \ i \ h \ j ) |
| \ / \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / \ / |
| \ h \ h \ e \ j \ i / |
| \ / \ / \ / \ / \ / |
| \ / \ / \ / \ / \ / |
| \ i \ g \ f \ h / |
| \ / \ / \ / \ / |
| \ / \ / \ / \ / |
| \ f \ e \ g / |
| \ / \ / \ / |
| \ / \ / \ / |
| \ g \ f / |
| \ / \ / |
| \ / \ / |
| \ e / |
| \ / |
| \ / |
| v |
| |
o-------------------------------------------------o
</pre>
|}
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:
# <math>\operatorname{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>x \cdot y.</math>
# <math>\operatorname{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>y \cdot x.</math>
<pre>
In (1), we consider the effects of each x in its
In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
practical bearing on contexts of the form <_, y>,