I just discovered that in the world of quandles the algebraic condition which corresponds to what I call “the shuffle trick” (or presentation or the video of this presentation at min. 45:00) has the name “medial” or “entropic”. In this language, what I can prove is that **entropic emergent algebras are affine spaces**, structure which I use in the Pure See to decorate chemlambda or directed interaction combinators graphs. Recall that the existence of the shuffle trick is what I argue shows that (at least the multiplicative part of) linear logic is actually commutative. There is a non-commutative version, which goes outside of using the shuffle trick, or algebraically the medial property, but it is so beautiful how things connect.

Looking forward to learn more about this, to see how my “emergent” part blends with the large literature on modes. I wouldn’t be very surprised if there exist semantics of linear logic built from modes.

I have a curious feeling by reading for example this article. I realize that I need some time time get used with the notations, even as a professional mathematician, but otherwise the article is full of intriguing words, like “distributed computation” and “theoretical biology”… Same dream.

**Update. **Here is a reasonably short, all-in-one description of the story.

Recall that an emergent algebra is a family of idempotent quasigroup operations over a set X, indexed by a parameter in a commutative group. The parameter is called “the scale”. The operations are called “dilations”.

I shall use in this post the letters a,b,c,… for scale parameters and x,y,z,w,… for the points in X.

I use a sort of polish notation here:

x y a is the dilation operation of x with y, at the scale a

The group operation over the scale parameters is denoted multiplicatively: ab, and the neutral element is 1. The inverse of a scale a is a’.

So a a’ = a’ a = 1.

We have also a (filter) 0 which is not element of the group of scales, but which is used for statements of the form:

“when a –> 0 the function f(a) –> F uniformly”.

To be able to say “uniformly” we need an uniformity over the set X. For example when X is a metric space, it will be the uniformity associated to the distance.

The algebraic axioms of emergent algebras are, with this notation:

(R1) x x a = x

(R2) x (x y a) b = x y (ab)

x y 1 = y

and as topological (or analytical) axioms: when a –> 0

x y a –> x uniformly wrt x,y in compact sets

define Delta_a (x,y,z) = x (x y a) (x z a) a’

then Delta_a (x,y,z)–> Delta(x,y,z) uniformly wrt x,y,z in compact sets

This is an emergent algebra.

An emergent algebra is linear if moreover we have this (scaled) distributivity

(LIN) x (y z a) b = (x y b) (x z b) a

An emergent algebra is medial, aka it satisfies the shuffle trick, if

(SHUF) (x u a) (y v a) b = (x y b) (u v b) a

With this we have:

- (SHUF) implies (LIN), so any medial emergent algebra is linear
- any linear emergent algebra comes from a conical group: for any

element x in X there is a group operation * on X, such that x is the neutral element of *, with y’ denoting the inverse of x wrt the group operation *, and such that

y z a = y * ( x (y^{-1} * z) a)

In particular if we satisfy some Lie group hypotheses (like in the solution of the Hilbert 5th problem) then the conical group is actually what a Carnot group.

Among Carnot groups, we have commutative ones, which are equivalent with vector spaces (suppose moreover that the group of scales is the multiplicative group over a field), and we have many other non-commutative ones, the most trivial example being the 2-nilpotent Heisenberg group.

- any medial emergent algebra is linear, by the previous result, but the conical group associated is commutative. Conversely, the emergent

algebra of a commutative conical group is medial.

So this is the relevance of the medial property (SHUF) for emergent algebras. Here is the relevance for chemlambda, directed interaction combinators or interaction combinators.

Let’s rewrite the operation

x y a = z

as the following statement:

from[a] x see[a] y as[a] z

If and only if the emergent algebra is medial there are other 5 medial emergent algebras, which are obtained by one of the other 5 permutations of 3 elements, given by any of the statements obtained from a permutation of (from, see, as).

This is not difficult to check for the medial emergent algebra of a vector space, because it just tells you that from

x y a = z

for a generic scale parameter a > 0

you can find other 5 expressions

y x (1-a) = z

x z (1/a) = y

y z (1/1-a) = x

z x (1 – 1/a) = y

z y (1 – 1/1-a) = x

which are all valid dilations!

Now, this can be used to decorate the nodes of chemlambda, directed interaction combinators (i.e. what I call dirIC) and by consequence interaction combinators.

The decorations are given in the Pure See draft

https://mbuliga.github.io/quinegraphs/puresee.html

The 6 nodes of chemlambda or dirIC are named,

D (dilation)

L (lambda)

A (application)

FI (a sort of fan-in)

FOE ( a sort of fan-out)

FOX (another sort of fanout)

To these, we may add

FO (fan-out) which is decorated with x,x,x

FIN (fan-in) which is decorated with x,x,x

which represent the fact that by (R1) x x a = x for any scale parameter a

chemlambda uses L, A, FI, FOE, FO nodes only.

dirIC uses L, A, FI, FOE nodes only.

This decoration of nodes which I just described has the property that any rewrite (of chemlambda or dirIC) we take, be it a beta-like

rewrite (i.e. like in the graphical beta reduction), or a DIST rewrite (as those used for duplication), if we decorate the LHS and RHS of the rewrite according to the rules explained, then we can always show that this rewrite is “emergent” (i.e. obtained from some scale parameters –>0) from a sequence of rewrites involving only (R1), (R2), and (SHUF).

Conversely, if we decorate chemlambda, dirIC graphs according to the rules just mentioned, we can prove (SHUF), in the sense that we can prove that the decoration has to have the (SHUF) property. This is done in chemlambda by the “shuffle trick”, which is a sequence of two rewrites involving 3 nodes.

Therefore (SHUF) is necessary and sufficient in the context,

Going back to the emergent algebras only part, there is more:

- we can express (LIN) as y z a = x ((x y b) (x z b) a) b’ , which is equivalent with

z = y (x ((x y b) (x z b) a) b’) a’

If we have a distance, say d, then

d(z, y (x ((x y b) (x z b) a) b’) a’ )

measures the difference from having the LIM property, for a generic emergent algebra. This is, I argue, related to curvature!

Likewise, if we have (LIN) then we can define a measure of the difference from having (SHUF). This is, if you compute, equal to the Lie bracket in the conical group!

Conclusion. All in all we have the following formalisms, in a somehow decreasing order of generality:

- the formalism of emergent algebras, which can be turned into a graph rewriting formalism over decorated graphs (with nodes and edges which are decorated) and with graph rewrites which take into account the decorations.
- less general is the formalism of linear emergent algebras, which can be used to (or are compatible with) a pure graph rewriting formalism over (non-planar) diagrams of tangles, because (LIN) in this case is the rewrite (R3). It is not known if this graph rewriting formalism is Turing complete though, or more precisely if there is some natural correspondence with Interaction Combinators, say. The purpose of the Zip Slip Smash formalism is to provide this correspondence, where we add to the Reidemeister rewrites some rewiring rewrites in order to achieve this.
- even less general is the formalism of medial emergent algebras, which turns out to be capable to serve as a semantics for dirIC, thus for the Interaction Combinators.

Therefore, the image is now, in increasing order of generality:

multiplicative linear logic < knot theory < differential calculus in emergent algebras

The interest would be to understand the implications of these inclusions, for example to provide versions of linear logic which are not commutative, but they are still LINear, therefore at the level of (LIN), not only (SHUF). Also, to decrease the differential calculus from the most general level to the level of (LIN), but not (SHUF) (where it is the usual one known to anybody), so that we can compare it on common ground with the linear logic, in a natural, unforced, less naive way.

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