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Directory:Jon Awbrey/Papers/Cactus Language (view source)
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The notation is a little bit awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that <math>\operatorname{E}_{ij}</math> has the effect of transforming each proposition <math>f : U \to \mathbb{B}</math> into a proposition <math>f^\prime : U \to \mathbb{B}.</math> As it happens, the action of each <math>\operatorname{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \operatorname{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\operatorname{T}_{00}, \operatorname{T}_{01}, \operatorname{T}_{10}, \operatorname{T}_{11},</math> to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:
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<pre>
<pre>
o----------o----------o----------o----------o----------o
o----------o----------o----------o----------o----------o
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o----------o----------o----------o----------o----------o
o----------o----------o----------o----------o----------o
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<pre>
It happens that there are just two possible groups of 4 elements.
It happens that there are just two possible groups of 4 elements.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.