[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 24

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Cybernetics List, Peirce List,

An iconic mapping, that gets formalized in mathematical terms as a "morphism",
is said to be a "structure-preserving map".  This does not mean that all of
the structure of the source domain is preserved in the map "images" of the
target domain, but only 'some' of the structure, that is, specific types
of relation that are defined among the elements of the source and target,
respectively.

For example, let's start with the archetype of all morphisms,
namely, a "linear function" or a "linear mapping" f : X -> Y.

To say that the function f is "linear" is to say that we have
already got in mind a couple of relations on X and Y that have
forms roughly analogous to "addition tables", so let's signify
their operation by means of the symbols "#" for "addition in X"
and "+" for "addition in Y".

More exactly, the use of "#" refers to a 3-adic relation L_X c X x X x X
that licenses the formula "a # b = c" just when <a, b, c> is in L_X, and
the use of "+" refers to a 3-adic relation L_Y c Y x Y x Y that licenses
the formula "p + q = r" just when <p, q, r> is in L_Y.

In this setting, the mapping f : X -> Y is said to be "linear",
and to "preserve" the structure of L_X in the structure of L_Y,
if and only if f(a # b) = f(a) + f(b), for all pairs a, b in X.
In other words, f "distributes" over the additions # to +, just
as if it were a form of multiplication, like m(a + b) = ma + mb.

Writing this more directly in terms of the 3-adic relations L_X and L_Y
instead of via their operation symbols, we would say that f : X -> Y is
linear with regard to L_X and L_Y if and only if <a, b, c> being in the
relation L_X determines that its map image <f(a), f(b), f(c)> be in L_Y.
To see this, observe that <a, b, c> being in L_X implies that c = a # b,
and <f(a), f(b), f(c)> being in L_Y implies that f(c) = f(a) + f(b), so
we have that f(a # b) = f(c) = f(a) + f(b), and the two notions are one.

The idea of mappings that preserve 3-adic relations should ring a few bells here.

Jon Awbrey

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