M Theory Lesson 217

As far as sets go, there are many cardinalities between $2 = | \{ 0,1 \} |$ and $\aleph_{1} = | \mathbb{C} |$. It is probably easier to start with 2 and work upwards, since Chu spaces for two truth values cover so much ground. Recall that a one dimensional truth valued matrix used the field with one element, called 1. This is combined with a zero in Set to form the truth set.

In discrete Fourier transforms, matrices have entries in the set $\{ 1, \omega, \cdots , \omega^{n-1} \} \cup \{ 0 \}$, which has $n + 1$ elements. That is, the number of truth values is the same as the number of MUB bases for dimension $n$, at least as long as $n$ is a prime power. Perhaps there is a good reason for this, especially since one of Pratt's basic examples is the $2 \times 2$ spinor. Anyway, Fourier operators with these restricted entries are a long way from requiring the whole set of complex numbers.

Returning to two point spaces: the Sierpinski space has one closed point and hence three open sets, namely the empty set, the set $\{ o \}$ and the set $\{ o , c \}$. The Chu matrix should therefore be a $2 \times 3$ matrix looking like

0 1 1
0 0 1

Now the really cool thing about the Chu calculator is that we can multiply Chu matrices together and then copy and paste the results into Blogger! Multiplying two Sierpinski spaces together we get the Chu matrix

011111
000111
001011
000001

OK, so that was multiplication. Hitting the addition button instead, we get

000111111
000000111
011011011
001001001

Multiplying the $3 \times 3$ matrix $(231)$ by itself we get

000011
001100
110000
010100
100001
001010
101000
010010
000101

Now try the Pauli swap matrix yourself. Then try a few more higher dimensional 1-circulants, and see how duality swaps them amongst themselves. What fun! And, yes, it has been snowing today.