This is part of the Tutorial:Graphic lambda calculus.

Here are described the moves related to emergent algebras. This moves involve the gates , and the termination gate.

See the post “3D crossings in emergent algebras” in order to understand the relation with knot diagram crossings and with Reidemeister moves for oriented knot diagrams (for those I use the notations from the paper “Minimal generating sets of Reidemeister moves“, by Michael Polyak only that I use the letter “R” from “Reidemeister” instead of “” used by Polyak).

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We have an abelian group with operation denoted multiplicatively and with neutral element denoted by . Thus, for any their product is , the inverse of is denoted by and we have the identities: , .

For each we have an associated gate .

is described in the following figure**The move R1a**

The reason for calling this move R1a is that is related to the move R1a from Polyak paper (also to R1d). In the papers on emergent algebras this move is called R1, that is “Reidemeister move 1”.

is this:**The move R1b**

This move is related to the move R1b from Polyak (and also to R1c). **This move does not appear in relation with general emergent algebras, it is true only for a special subclass of them, namely uniform idempotent quasigroups.**

is the following:**The move R2**

We shall see that this is related to all Reidemeister 2 moves (using also CO-ASSOC, CO-COMM, and LOCAL PRUNING).

The notation comes from the rule (ext2) from lambda-Scale calculus. In emergent algebra language it means that the emergent algebra operation indexed by the neutral element of is the trivial operation .**The move ext2.**

(but for this LOCAL PRUNING is also used).

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Marius, this is starting to look like Feynman diagrams!

Thanks, I shall look if there is a sector for Feynman diagrams in graphic lambda. Any way, it already proved to contain lambda calculus (say computation), differential calculus and knot diagrams.

Indeed, you may be right, a starting point could be section 2 from K. Yeats, Growth estimates for Dyson-Schwinger equations, arXiv:0810.2249 [math-ph] . It looks like a rather big task for the moment, but it might become easier with two added ingredients: help from others and time to think.