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Book: InterpretingCayley-DicksonConstruction

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Interpret the Cayley-Dickson construction.


Given a division algebra {$A$} with conjugates, define a new division algebra {$A\oplus A$} with the following product on the ordered pairs:


Define the conjugate as follows:

{$(a,b)^* = (a^*,-b)$}

The conjugate links an undistinguished opposite (one of two conjugates) with a distinguished opposite (positive or negative). Calculating we get {$((a,b),(c,d))^* = ((a,b)^*,-(c,d))=((a^*,-b),(-c,-d))$}.

In general, given a term {$(a_1,a_2,a_3,\cdots, a_{2^{n}}) \in \bigoplus_{i=1}^{2^n}A$} we have

{$$(a_1,a_2,a_3,\cdots, a_{2^{n}})^*=({a_1}^*,-a_2,-a_3,\cdots,-a_{2^{n}})$$}

Thus conjugates are relative and they reference an internal structural distinction, whereas positive and negative terms exhibit an absolute, external distinction. This relates to observation and measurement. It also brings to mind the notion of a head-driven phrase structure grammar. Ambiguity is maintained in the head. Everything subsequently explicitly supports the existing ambiguity.

Note that there are four dualities:

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