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Lefty wrote:
Snip >you can easily say
> that 1 = 7-6 = 10-9, etc etc, and that these separate things are
> identical because it can be proven - snip
But you would be wrong. The 1 is an integer, whilst 6,7,9 & 10 are
natural numbers; the symbol "-" is an ill defined conflation of two
concepts (negate & subtract). The correct statement is an equivalence
relationship 1 ~ {7,6} ~{10,9} (van der Waerden, Algebra, 1970,
Springer reprint 2003). Negation is an axiom (Axiom 4a, Allenby, Rings,
Fields & Groups, Blackie 1971) and other axioms give other algebras.
Subtraction is a difficult concept (Smith & Romanskaya, Post Modern
Algebra p111).
My Hoop algebras
(http://library.wolfram.com/infocenter/mathsource/4894) subsume many
"conservative partial-fraction division algebras", (including Real,
Complex, Quaternion, Octonion, Clifford, Davenport, Dirac, Lie, Pauli &
Wedge algebras) by starting with unsigned continuous "Primal" numbers,
Moufang Loop multiplication/division of sets of primal coefficients,
and "generalized signs" based on different roots of unity. The Hoop
algebra of 12 coefficients, "Dozal", has many analogies with particle
physics:-
1 deBroglie-wave-like multi-phase sinusoids, including quark-like
ternary symmetry
2 conserved Planck-area-like magnitudes
3 sub- and super-symmetries
4 Spin & half spin quantum operators - etc.
It may eventually provide the analogy that you seek.
Roger Beresford.
"He said true things, but called them by wrong names." (R. Browning.
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