# Emergent algebra moves R1a, R1b, R2 and ext2

This is part of the Tutorial:Graphic lambda calculus.

Here are described the moves related to emergent algebras. This moves involve the gates $\bar{\varepsilon}$, $\Upsilon$ 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 “$\Omega$” used by Polyak).

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We have an abelian group $\Gamma$ with operation denoted multiplicatively and with neutral element denoted by $1$. Thus, for any $\varepsilon, \mu = \mu \varepsilon \in \Gamma$ their product is $\varepsilon \mu \in \Gamma$, the inverse of $\varepsilon$ is denoted by $\varepsilon^{-1}$ and we have the identities: $1 \varepsilon = \varepsilon$ , $\varepsilon \varepsilon^{-1} = 1$.

For each $\varepsilon \in \Gamma$ we have an associated gate $\bar{\varepsilon}$.

• The move R1a is described in the following figure

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”.

• The move R1b is this:

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.

• The move R2 is the following:

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

• The move ext2. 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 $\Gamma$ is the trivial operation $xy = y$.

(but for this LOCAL PRUNING is also used).

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