1. The lambda epsilon calculus was introduced in arXiv:1205.0139 [cs.LO] “-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 -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 -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 -Scale there are two operations, namely the abstraction and a operation, one for every coefficient in a commutative group. The application operation from lambda calculus comes as a composite of these two fundamental operations. Moreover, the 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 -Scale (see the paper, definition 2.2).
Later, in graphic lambda calculus, I renounced at the operation of -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 operation of -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 operation from -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 . 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 , like this:
which is very nice, because we obtain the crossing macro of emergent algebras.
Finally, in the case when , 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.