Graph3D[GeneralizedGridGraph[{4"Directed",4"Directed",4"Directed"}]]

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## Projective Space

Projective Space

#### Metric in branch space

Metric in branch space

Minimum number of updates to derive both points

(i.e. its the common ancestor distance in the tree)

Finding that common ancestor distance might be computationally hard

---> geometric group theory

(i.e. its the common ancestor distance in the tree)

Finding that common ancestor distance might be computationally hard

---> geometric group theory

TransitiveClosureGraph[GridGraph[{5,5}]]

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TransitiveReductionGraph

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TransitiveReductionGraph,VertexLabelsAutomatic

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Table[TransitiveReductionGraph[MultiwaySystem[{"A""AB","B""A"},"A",t,"BranchialGraph"]],{t,2,5}]

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#### Projective space

Projective space

Modding out the metric by a constant

Tuples[Range[3],2]

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{{1,1},{1,2},{1,3},{2,1},{2,2},{2,3},{3,1},{3,2},{3,3}}

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If these were critical pairs, this must define a transitive binary relation......

What are the transitive binary relations on n elements?

https://math.stackexchange.com/questions/243773/amount-of-transitive-relations-on-a-finite-set

https://oeis.org/A006905

https://oeis.org/A000798/a000798_12.pdf

[ Quantum states are rays in Hilbert space ]

#### <Transitivity is discrete rescaling>

<Transitivity is discrete rescaling>