A simple explanation with types of the hexagonal moves of projective spaces

This post continues from the previous  A beautiful move in projective spaces    .

I call those “beautiful” moves hexagonal. So, how many hexagonal moves are?

The answer is 3.

In the post  Axioms for projective conical spaces (towards qubits II) I give 4 moves and I mention that I discard other two moves, because  don’t see their use in generalized projective geometry.

I was wrong, because in the usual projective geometry these two moves are discarded because they can be deduced from the barycentric move (which makes the generalized, “non-commutative”, projective geometry, into the usual one, in the same way as it makes the non-commutative affine geometry into the usual one). You really have to read the linked posts (and probably also the linked articles) in order to understand these precise statements. (So this is a kind of a filter for those with long attention span.)

The idea of using two types is natural: indeed, instead of using dual spaces, we use two types,  say “a” and “x”, and the decoration rule of the 4-valent dilation nodes:  two of the input arrows and the output arrow are decorated with the same type and the remaining input arrow is decorated with the other type. [ added: the “remaining input arrow” is always the one which points to the center of the circle which denoted the dilation node.]

By looking at the hexagonal moves from the last post, we see that there are only three ways of decorating the common part of all diagrams with types.  This gives 3 hexagonal moves.

This is explained in the next figure (click on it to make it bigger).

new_proj_7

We see that some arrows are decorated with one type (arbitrarily called “x”) and other arrows are decorated, apparently with columns of 3 types. In fact, that should be read as 3 possibilities, which correspond to taking from each column the first, the second or the third element.

The choice corresponding to the first element of each column corresponds to the neglected hexagonal move, say (PG3). (Can you draw it? is easy!)

The choice corresponding to the second element of each column corresponds to (PG2).

The choice corresponding to the third element of each column corresponds to (PG1).

Remarking that the rule of decoration with types is symmetric with the switch between the types “a” and “x”, this corresponds to the 6 moves of generalized projective geometry.

In conclusion, by selecting only those graphs (and moves) which can be decorated in the mentioned way with two types, we get the moves of generalized projective geometry.

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