Better than extended beta move

Instead of the extended graphic beta move (proposed here and declared still in beta version in the graphic lambda calculus tutorial) is to couple the Reidemeister 2  move R2 with the graphic beta move. Here is how it can be done.

Let us define, for any scale parameter , the following versions of the application gate and abstraction gate:

Remark that when we can recover the usual application gate

and the usual abstraction gate

The graphic beta move and the move R2 can be coupled into the following nice move:

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This will be used for making clear where the space is encoded in graphic lambda calculus and how.  Of course, it has to do with emergent algebras, seen in graphic lambda calculus. If you want to get ahead of explanations and figure out by yourself then the following posts will help:

• Dictionary from emergent algebra to graphic lambda calculus (I)
• Dictionary from emergent algebra to graphic lambda calculus (II)
• Dictionary from emergent algebra to graphic lambda calculus (III)

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(in case you see adds here: they have nothing to do with me or with this blog)