Lambda scale in graphic lambda calculus and diagram crossings

Background:

1. The lambda epsilon calculus was introduced in  arXiv:1205.0139 [cs.LO]    “$\lambda$-Scale, a lambda calculus for spaces with dilations”. Further, graphic lambda calculus was introduced in arXiv:1207.0332 [cs.LO]  “Local and global moves on locally planar trivalent graphs, lambda calculus and $\lambda$-Scale”.

2.   In the post “3D crossings in emergent algebras” I noticed a very intriguing resemblance between crossings in emergent algebras (as encoded in graphic lambda) and crossings macros in graphic lambda. More specifically, here is  the encoding of a crossing in emergent algebras:

which is very much like the crossing macro

Likewise, here is the other crossing in emergent algebras:

and its “twin” crossing macro

Their behaviour differs only with respect to the Reidemeister 3 move, which holds only in self-distributive (or “linear”) emergent algebras.

Question:   Are these crossing macros related in graphic lambda calculus?

Answer: Yes, through an idea introduced in the $\lambda$-Scale paper (which is actually the first proposal of a calculus for emergent algebra, later transformed into “graphic lambda calculus”).

Here is the explanation. In $\lambda$-Scale there are two operations, namely the $\lambda$ abstraction and a $\varepsilon$ operation, one for every coefficient $\varepsilon$ in a commutative group.  The application operation from lambda calculus comes as a composite of these two fundamental operations. Moreover, the $\varepsilon$ operation IS NOT THE SAME AS the operation from emergent algebras. The emergent algebra operation is also defined as a composite of the two fundamental operations of $\lambda$-Scale (see the paper, definition 2.2).

Later, in graphic lambda calculus, I renounced at the $\varepsilon$ operation of $\lambda$-Scale, replacing it with the operation from emergent algebras. To be clear, I used implicitly proposition 3.2 from the mentioned paper (which has a slightly misleading figure attached), which gives a description of the fundamental $\varepsilon$ operation of $\lambda$-Scale calculus in terms of the abstraction operation, the application operation  and the emergent algebra operation.

This turns out to be the key of the explanation I am writing about.

I shall define now a macro in graphic lambda calculus (inspired by the said proposition 3.2) which corresponds to the fundamental $\varepsilon$ operation from $\lambda$-Scale calculus. Here is it:

Let us play with this macro, by using it in a graph almost similar with the one where the graphic beta move applies:

Let us consider the particular case $\varepsilon = 1$. By using the (ext2) move and then local pruning we get the following graph:

I shall now apply the graphic beta move for the general $\varepsilon$, like this:

which is very nice, because we obtain the crossing macro of emergent algebras.

Finally, in the case when $\varepsilon = 1$, by an (ext2) move, followed by a local pruning pruning, we may transform the macro crossing from emergent algebra into this:

which ends the explanation.