For the shuffle trick see The illustrated shuffle trick… In a way, it’s also in Lafont’ Interaction Combinators article, in the semantics part.
It’s in the left part of Figure 14.
In chemlambda the pattern involves one FO node and two FOE nodes. In this pattern there is first a FO-FOE rewrite and then a FI-FOE one. After these rewrites we see that now we have a FOE instead of the FO node and two FO instead of the previous two FOE nodes. Also there is a swap of ports, like in the figure.
You can see it all in the linked post, an animation is this:
For previous posts about Lafont paper and relations with chemlambda see:
- A quine in Lafont’ Interaction combinators
- Universality of interaction combinators and chemical reactions
If the nodes FO and FOE were dilations of arbitrary coefficients a and b, in an emergent algebra, then the equivalent rewrite is possible if and only if we are in a vector space. (Hint: it implies linearity, which implies we are in a conical group, therefore we can use the particular form of dilations in the general shuffle trick and we obtain the commutativity of the group operation. The only commutative conical groups are vector spaces.)
In particular the em-convex axiom implies the shuffle trick, via theorem 8.9 from arXiv:1807.02058 . So the shuffle trick is a sign of commutativity. Hence chemlambda alone is still not general enough for my purposes.
You may find interesting the post Groups are numbers (1) . Together with the em-convex article, it may indeed be deduced that [the use of] one-parameter groups [in Gleason-Yamabe and Montgomery-Zippin] is analoguous to the Church encoding of naturals. One-parameter groups are numbers. The em-convex axiom could be weakened to the statement that 2 is invertible and we would still obtain theorem 8.9. So that’s when the vector space structure appears in the solution of the Hilbert 5th problem. But if you are in a general group with dilations, where only the “em” part of the em-convex rewrite system applies (together with some torsor rewrites, because it’s a group), then you can’t invert 2, or generally any other number than 1, so you get only a structure of conical group at the infinitesimal level. However naturals exist even there, but they are not related to one-parameter groups.