Unlimited detail is a sorting algorithm

This is a continuation of the post “Unlimited detail challenge…“.  I am still waiting for a real solution to emerge from the various intriguing ideas. Here I want to comment a bit about the kind of the algorithm UD might be.

Bruce Dell compared it with a search algorithm, more specifically compared it with a search algorithm for words. As everybody knows, words form an ordered set, with the lexicographic order.

Moreover, Dell explains that the thing of UD is to pick from the huge database (which might be in a server, while the algorithm runs on a computer elsewhere, thus the algorithm has to be in a sense an online algorithm) only the 3D atoms which are visible on the screen, then render only those atoms.

Therefore, imagine that we have a huge database of 3D atoms with some characteristics (like color) and a separate database made of the coordinates of these atoms. The UD algorithm solves a “sorting” problem in the second database. (I put “sorting” because there is no total order fully compatible with the solution of the ray-casting problem, in the sense that it remains the same when the POV is changed.) Once this “sorting” part is done, the algorithm asks for the characteristics of only those points and proceed to the rendering part, which is almost trivial.

By looking at sorting algorithms, it is then to be expected a time estimate of the kind  O((\log N)^{2}) (for given, fixed number of screen pixels), where N is the number of 3D atoms.

There are even sorting algorithms which are more efficient, like O(\log N), but they are more efficient in practice for really huge numbers of atoms , like 2^{6000}, as the AKS sorting.

So, the bottom line is this: think about UD as being a kind of sorting algorithm.

Emergent algebras as combinatory logic (Part II)

This post continues Emergent algebras as combinatory logic (Part I).  My purpose is to introduce the calculus standing behind Theorem 1 from the mentioned post.

We have seen (Definition 2) that there are approximate sum and difference operations associated to an emergent algebra. Let me add to them a third operation, namely the approximate inverse. For clarity I repeat here the Definition 2, supplementing it with the definition of the approximate inverse. This gives:

Definition 2′.   For any \varepsilon \in \Gamma we give the following names to several combinations of operations of emergent algebras:

  • the approximate sum operation is \Sigma^{x}_{\varepsilon} (u,v) = x \bullet_{\varepsilon} ((x \circ_{\varepsilon} u) \circ_{\varepsilon} v),
  • the approximate difference operation is \Delta^{x}_{\varepsilon} (u,v) = (x \circ_{\varepsilon} u) \bullet_{\varepsilon} (x \circ_{\varepsilon} v)
  • the approximate inverse operation is inv^{x}_{\varepsilon} u = (x \circ_{\varepsilon} u) \bullet_{\varepsilon} x.

The justification for these names comes from the explanations given in the post “The origin of emergent algebras (part II)“, where I discussed the sketch of a solution to the question “What makes the (metric)  tangent space (to a sub-riemannian regular manifold) a group?”, given by Bellaiche in the last two sections of his article  The tangent space in sub-riemannian geometry, in the book Sub-riemannian geometry, eds. A. Bellaiche, J.-J. Risler, Progress in Mathematics 144, Birkhauser 1996. We have seen there that the group operation (the noncommutative,  in principle, addition of vectors) can be seen as the limit of compositions of intrinsic dilations, as \varepsilon goes to 0. It is important that this limit exists and that it is uniform, according to Gromov’s hint.

Well,  with the notation \delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y, \delta^{x}_{\varepsilon^{-1}} y = x \bullet_{\varepsilon} y, it becomes clear, for example, that the composition of intrinsic dilations described in the figure from the post “The origin of emergent algebras (part II)” is nothing but the approximate sum from Definition 2′. (This is to say that formally, if we replace the emergent algebra operations with the respective intrinsic dilations, then the approximate sum operation \Sigma^{x}_{\varepsilon}(y,z)  appears as the red point E from the mentioned  figure. It is still left to prove that intrinsic dilations from regular sub-riemannian spaces give rise to emergent algebras, this was done in arXiv:0708.4298.)

We recognize therefore the two ingredients of Bellaiche’s solution into the definition of an emergent algebra:

  • approximate operations, which are just clever compositions of intrinsic dilations in the realm of sub-riemannian spaces, which
  • converge in a uniform way to the exact operations which give the algebraic structure of the tangent space.

Therefore, a rigorous formulation of Bellaiche’s solution is Theorem 1 from the previous post, provided that we extract,  from the long differential geometric work done by Bellaiche, only the part which is necessary for proving that intrinsic dilations produce an emergent algebra structure.

Nevertheless, Theorem 1 shows that the “emergence of operations” phenomenon is not at all specific to sub-riemannian geometry. In fact, once we get the idea of the right definition of approximate operations (from sub-riemannian geometry), we can simply try to prove the theorem by “abstract nonsense”, i.e. algebraically, with a dash of uniform convergence at the end.

For this we have to identify the algebraic relations which are satisfied by these approximate operations.  For example, is the approximate sum associative? is the approximate difference the inverse of the approximate sum? is the approximate inverse of an element the inverse with respect to the approximate sum? and so on. The answer to these questions is “approximately yes”.

It is clear that in order to find the right relations (approximate associativity and so on) between these approximate operations we need to reason in a more clear way. Just by looking at the expressions of the operations from Definition 2′, it is obvious that if we start with a brute force  “shut up and compute” approach  then we will end rather quickly with a mess of parantheses and coefficients. There has to be a more easy way to deal with those approximate operations than brute force.

The way I have found has to do with a graphical representation of these operations, a way which eventually led me to graphic lambda calculus. This is for next time.

Stephane Hessel

“Stéphane Frédéric Hessel (20 October 1917 – 27 February 2013) was a diplomat, ambassador, writer, concentration camp survivor, former French Resistance fighter and BCRA agent. Born German, he became a naturalised French citizen in 1939. He participated in the editing of the Universal Declaration of Human Rights of 1948. In 2011, he was named by Foreign Policy magazine to its list of top global thinkers. “[wiki dixit]

Hessel is the author of Indignez-vous! (french version, english version). In my mind, he stands near Aaron Swartz, like a kind of symbolic couple grandparent and grandson. This type of affinity, which jumps over a generation, seems to be a modern phenomenon. I began to recognize it by reading the very interesting book (in french, but maybe there are also english translations) ” Génération 69 : Les trentenaires ne vous disent pas merci” by Laurent Guimier and Nicolas Charbonneau.


Seven years forecast

  1. In seven years there will be no essential difference between comments on articles and peer-reviews. [28.02.2013 addition by Phillip Lord: “Blind peer-review will die, and open peer-review will take it’s place.” ]
  2. In seven years there will be semantic means of definition of plagiarism and, as a consequence, a significant percentage of today’s articles will qualify as recycled crap,
  3. In seven years there will be popularity contests and evaluations based on the popularity of the  authors as measured by their impact on the web,
  4. In seven years the best universities will gamify the teaching process,
  5. In seven years all  successful changes of the process of dissemination of knowledge will turn out to be among those born from private initiatives,
  6. In seven years large research collaborations of mathematicians will be regarded as normal,
  7. In seven years most of the articles which are now under the lock of the copyright belonging to the publisher will be seen as vanity publication and their most important use will be as data for programs of massive extraction of semantic content.

From combinators and zippers to interaction graphs (part I)

In the post Combinators and zippers I gave a description of the SKI combinators and their fundamental relations in terms of zippers.  See the tutorial page on graphic lambda calculus for background.

As a conclusion of the post  on combinators and zippers, let me mention that the I and K combinators appear as “half” of the 1-zipper and 2-zipper respectively, with some wires and termination gates added. The combinator S appears as a half of the 3-zipper with a “diamond” added, which in fact serves to generate more zippers when composed with other combinators.

Looks like gibberish? Here is for example the relation SKK \rightarrow I, as deduced in graphic lambda calculus, by using zippers:


Looks like I did it without using GLOBAL FAN-OUT! The point is that we may transform combinatory logic into a calculus with zippers, half-zippers (see how the left half zipper from the expression of K, made by two \lambda gates, couples to form a 2-zipper with the right-half zipper made by two \curlywedge gates from the “diamond”) and some wires and termination gates.

Here comes the more interesting part: could we do it with zippers written in terms of emergent algebras and by using the dual of the graphic beta move instead?

I shall first explain what a zipper in emergent algebra is. Look at the next figure to see one:


This is 3-zipper written with the help of the emergent algebra crossing macro. It unzips by using three \beta^{*}(\varepsilon) moves.

The problem with this zipper is that it is not at all clear what a half of it might be. Related to this is the fact that we can “unzip” independently each of the parts of the zipper, there is no order of zipping-unzipping, because we already have all the patterns needed to apply the dual of the graphic beta move. This is a different situation, compared to the one of the usual zipper. In the usual zipper case, that zipper is in fact made by two halves, one containing the \lambda gates, the other half containing the \curlywedge  gates; there is only one place where we can apply the graphic beta move and after the application of this move appears a new pattern proper for the application of the next graphic beta move and so on, until the zipper becomes unzipped.

In the extreme case of the 1-zipper, which is nothing but the pattern for the application of the graphic beta move, we can identify the \lambda  gate with the left half of the zipper and the \curlywedge gate with the right half of the zipper. On the contrary, we can’t do anything comparable to this for the 1-zipper from emergent algebras.

A solution which could get us out of this problem (and therefore one which would allow to do combinatory logic in the realm of emergent algebra) is to introduce “interaction graphs”, which form a new sector of the graphic lambda calculus.  These interaction graphs look (superficially) a bit alike Feynman trivalent diagrams, with “interaction  nodes” decorated either with \lambda or with \curlywedge, and wires corresponding to “particles” \Upsilon or \varepsilon. This is for the next time, I have to play a bit more with those before showing them.

The origin of emergent algebras (part II)

I continue from the post “The origin of emergent algebras“, which revolves around the last sections of Bellaiche paper The tangent space in sub-riemannian geometry, in the book Sub-riemannian geometry, eds. A. Bellaiche, J.-J. Risler, Progress in Mathematics 144, Birkhauser 1996.

In this post we shall see how Bellaiche proposes to extract the algebraic structure of the metric  tangent space T_{p}M at a point p \in M, where M is a regular sub-riemannian manifold. Remember that the metric tangent space is defined up to arbitrary isometries fixing one point, as the limit in the Gromov-Hausdorff topology over isometry classes of compact pointed metric spaces

[T_{p} M, d^{p}, p] = \lim_{\varepsilon \rightarrow 0} [\bar{B}(p, \varepsilon), \frac{1}{\varepsilon} d, p]

where [X, d, p] is the isometry class of the compact  metric space (X,d) with a marked point p \in X. (Bellaiche’s notation is less precise but his previous explanations clarify that his relations (83), (84) are meaning exactly what I have written above).

A very important point is that moreover, this convergence is uniform with respect to the point p \in M. According to Gromov’s hint mentioned  by  Bellaiche, this is the central point of the matter. By using this and the structure of the trivial pair groupoid M \times M, Bellaiche proposes to recover the Carnot group algebraic structure of T_{p}M.

From this point on I shall pass to a personal interpretation of the  section 8.2 “A purely metric derivation of the group structure in T_{p}M for regular p” of Bellaiche article. [We don’t have to worry about “regular” points because I already supposed that the manifold is “regular”, although Bellaiche’s results are more general, in the sense that they apply also to sub-riemannian manifolds which are not regular, like the Grushin plane.]

In order to exploit the limit in the sense of Gromov-Hausdorff, he needs first an embodiment of the abstract isometry classes of pointed metric spaces. More precisely, for any \varepsilon > 0 (but sufficiently small), he uses a function denoted by \phi_{x}, which he states that it is defined on T_{x} M with values in M. But doing so would be contradictory with the goal of constructing the tangent space from the structure of the trivial pair groupoid and dilations. For the moment there is no intrinsic meaning of T_{x} M, although there is one from differential geometry, which we are not allowed to use, because it is not intrinsic to the problem.  Nevertheless, Bellaiche already has the functions \phi_{x}, by way of his lengthy proof (but up to date the best proof) of the existence of adapted coordinates. For a detailed discussion see my article “Dilatation structures in sub-riemannian geometry” arXiv:0708.4298.

Moreover, later he mentions “dilations”, but which ones? The natural dilations he has from the vector space structure of the tangent space in the usual differential geometric sense? This would have no meaning, when compared to his assertion that the structure of a Carnot group of the metric tangent space is concealed in dilations.  The correct choice is again to use his adapted coordinate systems and use intrinsic dilations.  In fewer words, what Bellaiche probably means is that his functions \phi_{x} are also decorated with the scale  parameter \varepsilon >0, so they should deserve the better notation \phi_{\varepsilon}^{x},  and that these functions behave like dilations.

A natural alternative to Bellaiche’s proposal would be to use an embodiment of the isometry class [\bar{B}(x, \varepsilon), \frac{1}{\varepsilon} d, x] on the space M, instead of the differential geometric tangent space T_{x}M.  With this choice, what Bellaiche is saying is that we should consider dilation like functions \delta^{x}_{\varepsilon} defined locally from M to M such that:

  • they preserve the point x (which will become the “0” of the metric tangent space): \delta^{x}_{\varepsilon} x = x
  • they form a one-parameter group with respect to the scale: \delta^{x}_{\varepsilon} \delta^{x}_{\mu} y = \delta^{x}_{\varepsilon \mu} y and \delta^{x}_{1} y = y,
  • for any y, z at a finite distance from x (measured with the sub-riemannian distance d, more specifically such that  d(x,y), d(x,z) \leq 1) we have

d^{x}(y,z) = \frac{1}{\varepsilon} d( \delta^{x}_{\varepsilon} y, \delta^{x}_{\varepsilon}z) + O(\varepsilon)

where O (\varepsilon) is uniform w.r.t. (does not depend on) x, y , z in compact sets.

Moreover, we have to keep in mind that the “dilation”  \delta^{x}_{\varepsilon} is defined only locally, so we have to avoid to go far from x, for example we have to avoid to apply the dilation for \varepsilon very big to points at finite distance from x.

Again, the main thing to keep in mind is the uniformity assumption. The choice of the embodiment provided by “dilations” is not essential, we may take them otherwise as we please, with the condition that at the limit \varepsilon \rightarrow 0 certain combinations of dilations converge uniformly. This idea suggested by Bellaiche reflects the hint by Gromov.  In fact this is what is left from the idea of a manifold in the realm of sub-riemannian geometry  (because adapted coordinates cannot be used for building manifold structures, due to the fact that “local” and “infinitesimal” are not the same in sub-riemannian geometry, a thing rather easy to misunderstand until you get used to it).

Let me come back to Bellaiche reasoning, in the setting I just explained. His purpose is to construct the operation in the tangent space, i.e. the addition of vectors. Only that the addition has to recover the structure of a Carnot group, as proven by Bellaiche. This means that the addition is not a commutative, but a noncommutative  nilpotent operation.

OK, so we have the base point x \in M and two near points y and z, which are fixed. The problem is how to construct an intrinsic addition of y and z with respect to x. Let us denote by y +_{x} z the result we are seeking. (The link with the trivial pair groupoid is that we want to define an operation which takes (x,y) and (x,z) as input and spills (x, y+_{x} z) as output.)

The relevant figure is the following one, which is an improved version of the Figure 5, page 76 of Bellaiche paper.


Bellaiche’s recipe has to do with the points in blue. He says that first we have to go far from x, by dilating the point z w.r.t. the point x, with the coefficient \varepsilon^{-1}. Here \varepsilon is considered to be small (it will go to 0), therefore \varepsilon^{-1} is big.  The result is the blue point A. Then, we dilate (or rather contract) the point A  by the coefficient \varepsilon w.r.t. the point y. The result is the blue point B.

Bellaiche claims that when \varepsilon goes to 0 the point B converges to the sum y +_{x} z. Also, from this intrinsic definition of addition, all the other properties (Carnot group structure) of the operation may be deduced from the uniformity of this convergence. He does not give a proof of this fact.

The idea of Bellaiche is partially correct (in regards to the emergence of the algebraic properties of the operation from uniformity of the convergence of its definition) and partially wrong (this is not the correct definition of the operation). Let me start with the second part. The definition of the operation has the obvious default that it uses the point A which is far from x. This is in contradiction with the local character of the definition of the metric tangent space (and in contradiction with the local definition of dilations).  But he is wrong from interesting reasons, as we shall see.

Instead, a slightly different path could be followed, figured by the red points C, D, E. Indeed, instead of going far away first (the blue point A), then coming back at finite distance from x (the blue point B), we may first come close to x (by using  the red points C, D), then inflate the point D to finite distance from x and get the point E. The recipe is a bit more complicated, it involves three dilations instead of two, but I can prove that it works (and leads to the definition of dilation structures and later to the definition of emergent algebras).

The interesting part is that if we draw, as in the figure here,  the constructions in the euclidean plane, then we get E = B, so actually in this case there is no difference between the outcome of these constructions. At further examination this looks like an affine feature, right? But in fact this is true in non-affine situations, for example in the case of intrinsic dilations in Carnot groups, see the examples from the post “Emergent algebra as combinatory logic (part I)“.

Let’s think again about these dilations, which are central to our discussion, as being operations. We may change the notations like this:

\delta^{x}_{\varepsilon} y = x \circ_{\varepsilon} y

Then, it is easy to verify that the equality between the red point E and the blue point B is a consequence of the fact that in usual vector spaces (as well as in their non-commutative version, which are Carnot groups), the dilations, seen as operations, are self-distributive! That is why Bellaiche is actually right in his definition of the tangent space addition operation, provided that it is used only for self-distributive dilation operations. (But this choice limits the applications of his definition of addition operation only to Carnot groups).

Closing remark: I was sensible to these two last sections of Bellaiche’s paper because I was prepared by one of my previous obsessions, namely how to construct differentiability only from topological data.  This was the subject of my first paper, see the story told in the post “Topological substratum of the derivative“, there is still some mystery to it, see arXiv:0911.4619.

Emergent algebras as combinatory logic (Part I)

At some point this new thread I am starting now will meet the Towards qubits thread.

Definition 1.  Let \Gamma  be a commutative group with neutral element denoted by 1 and operation denoted multiplicatively. A \Gamma idempotent quasigroup is a set X endowed with a family of operations \circ_{\varepsilon}: X \times X \rightarrow X,  indexed by \varepsilon \in \Gamma, such that:

  1. For any \varepsilon \in \Gamma the pair (X, \circ_{\varepsilon}) is an idempotent quasigroup,
  2. The operation \circ_{1} is trivial: for any x,y \in X we have x \circ_{1} y = y,
  3. For any x, y \in X and any \varepsilon, \mu \in \Gamma we have:  x \circ_{\varepsilon} ( x \circ_{\mu} y) = x \circ_{\varepsilon \mu} y.

This definition may look strange, let me give some examples of \Gamma idempotent quasigroups.

Example 1.  Real vector spaces: let X be a real vector space, \Gamma = (0,+\infty) with multiplication of reals as operation. We define, for any \varepsilon > 0 the following quasigroup operation:

x \circ_{\varepsilon} y = (1-\varepsilon) x + \varepsilon y

These operations give to X the structure of a $(0,+\infty)$ idempotent quasigroup.  Notice that x \circ_{\varepsilon}y is the dilation based at x, of coefficient \varepsilon, applied to y.

Example 2. Complex vector spaces: if X is a complex vector space then we may take \Gamma = \mathbb{C}^{*} and we continue as previously, obtaining an example of a \mathbb{C}^{*} idempotent quasigroup.

Example 3.  Contractible groups: let G be a group endowed with a group morphism \phi: G \rightarrow G. Let \Gamma = \mathbb{Z} with the operation of addition of integers (thus we may adapt Definition 1 to this example by using “\varepsilon + \mu” instead of “\varepsilon \mu” and “0” instead of “1” as the name of the neutral element of \Gamma).  For any \varepsilon \in \mathbb{Z} let

x \circ_{\varepsilon} y = x \phi^{\varepsilon}(x^{-1} y)

This a \mathbb{Z} idempotent quasigroup. The most interesting case (relevant also for Definition 3 below) is the one when \phi is an uniformly contractive automorphism of the topological group G. The structure of these groups is an active exploration area, see for example arXiv:0704.3737 by  Helge Glockner   and the bibliography therein  (a fundamental result here is Siebert article Contractive automorphisms on locally compact groups, Mathematische Zeitschrift 1986, Volume 191, Issue 1, pp 73-90).  See also conical groups and relations between contractive and conical groups introduced in arXiv:0804.0135,  shortly explained in arXiv:1005.5031.

Example 4.  Carnot groups: these are a particular example of a conical group. The most trivial noncommutative Carnot group is the Heisenberg group.

Example 5. A group with an invertible self-mapping \phi: G \rightarrow G  such that \phi(e) =e, where e is the identity of the group G. In this case the construction from Example 3 works here as well because there is no need for \phi to be a group morphism.

Example 6. Local versions. We may accept that there is a way (definitely needing care to well formulate, but intuitively cleart) to define a local version of the notion of a \Gamma  idempotent quasigroup. With such a definition, for example, a convex subset of a real vector space gives a local $(0,+\infty)$ idempotent quasigroup (as in Example 1) and a neighbourhood of the identity of a topological group G, with an identity preserving, locally defined invertible self map (as in Example 5) gives a \mathbb{Z} local idempotent quasigroup.

Example 7. A particular case of Example 6, is a Lie group G with the operations  defined for any \varepsilon \in (0,+\infty) by

x \circ_{\varepsilon} y = x \exp ( \varepsilon \log (x^{-1} y) )

Example 8. A less symmetric example is the one of X being a riemannian manifold, with associated operations  defined for any \varepsilon \in (0,+\infty) by

x \circ_{\varepsilon}y = \exp_{x}( \varepsilon \log_{x}(y))

Example 9. More generally, any metric space with dilations  (introduced in  arXiv:math/0608536[MG] )  is a local idempotent quasigroup.

Example 10.  One parameter deformations of quandles. A quandle is a self-distributive quasigroup. Take now a one-parameter family of quandles (indexed by \varepsilon \in \Gamma) which satisfies moreover points 2. and 3. from Definition 1. What is interesting about this example is that quandles appear as decorations of knot diagrams, which are preserved by the Reidemeister moves.  At closer examination, examples 1-4 are all particular cases of one parameter quandle deformations!


I shall define now the operations of approximate sum and approximate difference associated to a \Gamma  idempotent quasigroup.

For any \varepsilon \in \Gamma, let use define x \bullet_{\varepsilon} y = x \circ_{\varepsilon^{-1}} y.

Definition 2.  The approximate sum operation is (for any \varepsilon \in \Gamma)

\Sigma_{\varepsilon}^{x}(y,z) = x \bullet_{\varepsilon} ( (x \circ_{\varepsilon} y) \circ_{\varepsilon} z)

The approximate difference operation is (for any \varepsilon \in \Gamma)

\Delta_{\varepsilon}^{x}(y,z) = (x \circ_{\varepsilon} y) \bullet_{\varepsilon} (x \circ_{\varepsilon} z)


Suppose now that X is a separable uniform space.  Let us suppose that the commutative group $\Gamma$ is a topological group endowed with an absolute, i.e. with an invariant topological filter, denoted by 0. We write \varepsilon \rightarrow 0 for a net in $\latex \Gamma$ which converges to the filter 0. The image to have in mind is \Gamma = (0, + \infty) with multiplication of reals as operation and with the filter 0 as the filter generated by sets (0, a) with a> 0. This filter is the restriction to the set (0,\infty) \subset \Gamma of the filter of  neighbourhoods of the number 0 \in \mathbb{R}.  Another example is \Gamma = \mathbb{Z} with addition of integers as operation, seen as a discrete topological group, with the absolute generated by sets \left\{ n \in \mathbb{Z} \, \, : \, \, n \leq M \right\} for all M \in \mathbb{Z}. For this example the neutral element (denoted by 1 in Definition 1) is the integer 0, therefore in this case we can change notations from multiplication to addition, 1 becomes 0, the absolute 0 becomes - \infty , and so on.

Definition 3. An emergent algebra (or uniform idempotent quasigroup) is a \Gamma idempotent quasigroup  X, as in Definition 1, which satisfies the following topological conditions:

  1. The family of operations \circ_{\varepsilon} is compactly contractive, i.e. for any compact set K \subset X, for any x \in K and for any open neighbourhood U of x, there is an open set A(K,U) \subset \Gamma which belongs to the absolute 0 such that for any u \in K and \varepsilon \in A(K,U) we have x \circ_{\varepsilon} u \in U.
  2. As \varepsilon \rightarrow 0 there exist the limits

\lim_{\varepsilon \rightarrow 0} \Sigma^{x}_{\varepsilon} (y,z) = \Sigma^{x} (y,z)  and \lim_{\varepsilon \rightarrow 0} \Delta^{x}_{\varepsilon} (y,z) = \Delta^{x} (y,z)

and moreover these limits are uniform with respect to x,y,z in compact sets.

The structure theorem of emergent algebras is the following:

Theorem 1.  Let X be  a \Gamma emergent algebra. Then for any x \in X the pair (X, \Sigma^{x}(\cdot, \cdot)) is a conical group.

In the next post on this subject I shall explain why this is true, in the language of graphic lambda calculus.