Approximate algebraic structures, emergent algebras

I updated and submitted for publication the paper “Emergent algebras“.

This is the first paper where emergent algebras appear. The subject is further developed in the paper “Braided spaces with dilations and sub-riemannian symmetric spaces“.

I strongly believe this is a very important notion, because it shows how  both the differential and algebraic realms  emerge naturally, from abstract nonsense. It is a “low tech” approach, meaning that I don’t use in the construction any “high tech” mathematical object, everything is growing from the grass.

One interesting fact, apart from the strange ideas of the paper (it is already verified that reading the paper algebraists will not understand easily the strength of the axiom concerning uniform convergence and analysts will not care enough about the occurrence of algebraic structure very much alike quandles and racks), is that an emergent algebra can also be seen as an approximate algebraic structure! But in a different sense than approximate groups.  The operations themselves are approximately associative, for example.

And my next question is: is this a really different notion of approximate algebraic structure than approximate groups? Or there is a way to see, for example, an approximate group (btw, why not an approximate symmetric space in the sense of Loos, whatever this could mean?) as an emergent algebra?

My hope is that the answer is YES.

UPDATE:   No, in fact there are reasons to think that there is a complementarity, there is a mathematical object standing over both, which may be called POSITIONAL SYSTEM, more soon, but see also this previous post of mine.

Here is the abstract of the paper:

“Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras.
Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example.
An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters.
Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property. “

On the difference of two Lipschitz functions defined on a Carnot group

Motivation for this post: the paper “Lipschitz and biLipschitz Maps on Carnot Groups” by William Meyerson. I don’t get it, even after several readings of the paper.

The proof of Fact 2.10 (page 10) starts by the statement that the difference of two Lipschitz functions is Lipschitz and the difference of two Pansu differentiable functions is differentiable.

Let us see: we have a Carnot group (which I shall assume is not commutative!) $G$ and two functions $f,g: U \subset G \rightarrow G$, where $U$ is an open set in $G$. (We may consider instead two Carnot groups $G$ and $H$ (both non commutative) and two functions $f,g: U \subset G \rightarrow H$.)

Denote by $h$ the difference of these functions: for any $x \in U$ $h(x) = f(x) (g(x))^{-1}$  (here the group operations  and inverses are denoted multiplicatively, thus if $G = \mathbb{R}^{n}$ then $h(x) = f(x) - g(x)$; but I shall suppose further that we work only in groups which are NOT commutative).

1.  Suppose $f$ and $g$ are Lipschitz with respect to the respective  CC left invariant distances (constructed from a choice of  euclidean norms on their respective left invariant distributions).   Is the function $h$ Lipschitz?

NO! Indeed, consider the Lipschitz functions $f(x) = x$, the identity function,  and $g(x) = u$ a constant function, with $u$ not in the center of $G$. Then $h$ is a right translation, notoriously NOT Lipschitz with respect to a CC left invariant distance.

2. Suppose instead that $f$ and $g$ are everywhere Pansu differentiable and let us compute the Pansu “finite difference”:

$(D_{\varepsilon} h )(x,u) = \delta_{\varepsilon^{-1}} ( h(x)^{-1} h(x \delta_{\varepsilon} u) )$

We get that $(D_{\varepsilon} h )(x,u)$ is the product w.r.t. the group operation of two terms: the first is the conjugation of the finite difference $(D_{\varepsilon} f )(x,u)$  by $\delta_{\varepsilon^{-1}} ( g(x) )$ and the second term is the finite difference   $(D_{\varepsilon} g^{-1} )(x,u)$.  (Here  $Inn(u)(v) = u v u^{-1}$ is the conjugation of $v$ by $u$ in the group $G$.)

Due to the non commutativity of the group operation, there should be some miracle in order for the finite difference of $h$ to converge, as $\varepsilon$ goes to zero.

We may take instead the sum of two differentiable functions, is it differentiable (in the sense of Pansu?). No, except in very particular situations,  because we can’t get rid of the conjugation, because the conjugation is not a Pansu differentiable function.