I think it is clear classic string theory is defined in Z2, if only because waves are 1 and nodes are zero.
If we assume the string waves and nodes are defined in Z2 as a deductive system, the fundamental is defined on the interval [0, 1] with the wave boundary condition 0, 1, 0 and the identity of the fundamental is [0, 1].
In "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea (Springer, 2007) the F2 = {0, 1} field is described, but I am having trouble with the way that multiplication and addition are defined.
I would like to know if this is the only way that addition and multiplication in the F2 field can be defined. Or, is it possible the way it is in the text is a natural representation of the string wave boundary that I don't get?
The book says these are the rules for F2:
0 + 0 = 1 + 1 = 0
0 + 1 = 1 + 0 = 1
0 x 0 = 0 x 1 = 1 x 0 = 0
1 x 1 = 1
But it seems more like it should be;:
1 + 1 = 1
1 x 1 = 1
The reason I think this is because if we add two waves we get a new mode. If we add two octaves we get a new octave. If we add two strings we get a new unit. The 1 + 1 = 1 equation describes the behavior of projective sets and other topologic sets like algebraic fields (and I think nullstellensatz). If two closed sets can form a closed union and intersection, then they make an arrow with a 1. Or just a 1.
I want Z2 to be the the principle ideals of the algebra of the string. The string is the union, intersection, and complimentaion of two disjoint sets (which are the pitch and position set defined on the string).
I think that since the boundary and matrices for [ Z2]n cannot add or multiply if the dimensions are not equivalent, this shows waves with different dimensions do not add.
The string has to be an integral domain and not the sum of an infinite series.
Sorry, I'm not enough of a physicist to follow the reasons you would like the operations in F2 to include 1+1 =1, but there is a simple algebraic reason this cannot happen from the point of view of fields. Each element of a field must have an additive inverse, so since 1 is in the field, there is an element -1 in the field such that 1 + (-1) = 0. If 1+1=1, then by adding -1 to both sides, you'd have 1=0, which is a contradiction (the assumption is that 0 and 1 are the distinct elements of the field). So there is no choice in this field: 1+1 must equal 0.
For instance, 1 octave + 1 octave = 1 octave.
The octave point is (1, 1) and the unit between (0, 0) is 1. Then the octave point + the octave interval add to zero.
In a Boolean atom we have the sides and diagonal of the square equal to 1. The triangle inequality a + b is less than or equal to c, becomes a + b = c. Since every distance is 1 every a = a2.
Tell me if this makes sense to you: The field over Z2 is formed by Tarski's set of all sentences where f(0, 0), f(1, 0), and f(1, 1) = 1 and f(1, 0) = 0. This is the propositional logic table for implication (if and only if).
Does that make a field F2 where the octave identity includes both 0 and 1 so multiplication and addition are the same.
Note that notes and intervals are atomic: pitch and not pitch. Dense: Between every two notes is an interval. Between every two intervals is a not. Meet and join make a lattice. A lattice has 2 algebraic and 1 logic sub lattice. 1 + 1 = 1 makes sense in projective sets. A projective set + projective set is a projective set. Topology + topology = topology. Algebraic field plus algebraic field is algebraic field.
Thank you for your answer.
No, sorry, most of this does not make sense to me. The bottom line is that there is no way you are going to make the two-element set F2 into a field with operations + and * that are the same.
If you are trying to model the notes in an octave via an interval on the real line and you want to identify notes that differ by an integer number of octaves as the same, the natural way to do that would seem to be the quotient group R/Z (the real numbers modulo the integers), which is isomorphic to the circle group S1. Note that this is only a group, though, not a field.
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