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.