[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 14

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The next few excursions in this series will provide
a scenic tour of various ideas in group theory that
will turn out to be of constant guidance in several
of the settings that are associated with our topic.

Let me return to Peirce's early papers on the algebra of relatives
to pick up the conventions that he used there, and then rewrite my
account of regular representations in a way that conforms to those.

Peirce expresses the action of an "elementary dual relative" like so:

| [Let] A:B be taken to denote
| the elementary relative which
| multiplied into B gives A.
|
| Peirce, 'Collected Papers', CP 3.123.

Peirce is well aware that it is not at all necessary to arrange the
elementary relatives of a relation into arrays, matrices, or tables,
but when he does so he tends to prefer organizing 2-adic relations
in the following manner:

   a:b  a:b  a:c

   b:a  b:b  b:c

   c:a  c:b  c:c

For example, suppose that we have the set X = {a, b, c},
together with the 2-adic relative term m = "marker for"
and the corresponding 2-adic relation M c X x X whose
general forms are suggested by the following matrix:

   M  =

   M_aa a:a  +  M_ab a:b  +  M_ac a:c  +

   M_ba b:a  +  M_bb b:b  +  M_bc b:c  +

   M_ca c:a  +  M_cb c:b  +  M_cc c:c

It has long been customary to omit the implicit plus signs
in these matrical displays, but I have restored them here
simply as a way of separating terms in this blancophage
web format.

For at least a little while, I will make explicit
the distinction between a "relative term" like m
and a "relation" like M c X x X, but it is best
to think of both of these entities as involving
different applications of the same information,
and so we might just as easily write this form:

   m  =

   m_aa a:a  +  m_ab a:b  +  m_ac a:c  +

   m_ba b:a  +  m_bb b:b  +  m_bc b:c  +

   m_ca c:a  +  m_cb c:b  +  m_cc c:c

By way of making up a concrete example,
let us say that M is given as follows:

   a is a marker for a

   a is a marker for b

   b is a marker for b

   b is a marker for c

   c is a marker for c

   c is a marker for a

In sum, we have this matrix:

   M  =

   1 a:a  +  1 a:b  +  0 a:c  +

   0 b:a  +  1 b:b  +  1 b:c  +

   1 c:a  +  0 c:b  +  1 c:c

I think that will serve to fix notation
and set up the remainder of the account.

Jon Awbrey

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