ONT Re: Differential Logic -- Series A
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note A19
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
To construct the regular representations of S_3,
we pick up from the data of its operation table:
Table 1. Symmetric Group S_3
| ^
| 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
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 L c G^3 is actually a function L : G x G -> G.
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 L : G x G -> G,
we can define a couple of substitution operators:
1. Sub(x, <_, y>) puts any specified x into
the empty slot of the rheme <_, y>, with
the effect of producing the saturated
rheme <x, y> that evaluates to xy.
2. Sub(x, <y, _>) puts any specified x into
the empty slot of the rheme <y, _>, with
the effect of producing the saturated
rheme <y, x> that evaluates to yx.
In (1), 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 xy, for y in G, all of which
is summarily notated as x = {(y : xy) : y in G}.
The pairs (y : xy) can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin. This produces
the regular ante-representation of S_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:j + i:h + j:i
g = e:g + f:e + g:f + h:i + i:j + j:h
h = e:h + f:i + g:j + h:e + i:f + j:g
i = e:i + f:j + g:h + h:g + i:e + j:f
j = e:j + f:h + g:i + h:f + i:g + j:e
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_3, like so:
e = e:e + f:f + g:g + h:h + i:i + j:j
f = e:f + f:g + g:e + h:i + i:j + j:h
g = e:g + f:e + g:f + h:j + i:h + j:i
h = e:h + f:j + g:i + h:e + i:g + j:f
i = e:i + f:h + g:j + h:f + i:e + j:g
j = e:j + f:i + g:h + h:g + i:f + j:e
If the ante-rep looks different from the post-rep,
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.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o