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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 21:15, 23 April 2008
, 21:15, 23 April 2008→Note 19: markup
j = e:j + f:h + g:i + h:f + i:g + j:e
j = e:j + f:h + g:i + h:f + i:g + j:e
</pre>
</pre>
In (2), we consider the effects of each ''x'' in its practical bearing on contexts of the form «''y'', _», as ''y'' ranges over ''G'', and the effects are such that ''x'' takes «''y'', _» into ''yx'', for ''y'' in ''G'', all of which is summarily notated as ''x'' = {(''y'' : ''yx'') : ''y'' in ''G''}. The pairs (''y'' : ''yx'') can be found by picking an ''x'' on the right margin of the group operation table and considering its effects on each ''y'' in turn as these run along the left margin. This generates the regular post-representation of ''S''<sub>3</sub>, like so:
<pre>
<pre>
e = e:e + f:f + g:g + h:h + i:i + j:j
e = e:e + f:f + g:g + h:h + i:i + j:j
j = e:j + f:i + g:h + h:g + i:f + j:e
j = e:j + f:i + g:h + h:g + i:f + j:e
</pre>
If the ante-rep looks different from the post-rep,
If the ante-rep looks different from the post-rep, it is just as it should be, as ''S''<sub>3</sub> 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.
it is just as it should be, as S_3 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.
===Note 20===
===Note 20===