Lambda scale in graphic lambda calculus and diagram crossings


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.

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