[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 18

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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 V_4, let us
write out as quickly as possible in "relative form" a minimal budget
of representations for the symmetric group on three letters, Sym(3).
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 "Algebra + Logic" papers.

Writing the permutations or substitutions of Sym {a, b, c}
in relative form generates what is generally thought of as
a "natural representation" of S_3.

   e  =  a:a + b:b + c:c

   f  =  a:c + b:a + c:b

   g  =  a:b + b:c + c:a

   h  =  a:a + b:c + c:b

   i  =  a:c + b:b + c:a

   j  =  a:b + b:a + c:c

I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as x:y
constitutes the turning of x into y.  It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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