Emergent algebras as combinatory logic (Part III)

I am continuing from Emergent algebras as combinatory logic (Part II).  In this post I am introducing the emergent algebra sector and we are starting to see how relations between approximate operations can be understood as computations.

Before starting the explanations, let me mention that  I think it has not been stressed enough that emergent algebras are not only about how  “exact” algebraic structures emerge from “approximate” ones, but also about the fact that emergent algebras come with their own notion of differentiability. Moreover, from the viewpoint of emergent algebras there is no essential difference between the emergence of exact operations from approximate ones and differentiation! But this is for a future post, to come soon.

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In the next definition I shall introduce some graphs in GRAPH which I claim they correspond to the approximate operations from Definition 2′.

Definition 3. For any \varepsilon \in \Gamma , the following graphs in GRAPH are introduced:

  • the approximate sum graph \Sigma_{\varepsilon}

approx_op_2

  • the approximate difference graph \Delta_{\varepsilon}

approx_op_3

  • the approximate inverse graph inv_{\varepsilon}

approx_op_4

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Let A be a set of symbols a, b, c, ....   (These symbols will play the role of  scale parameters going to 0.) With A and with the abelian group \Gamma we construct a larger abelian group, call it \bar{\Gamma}, which is generated by A and by \Gamma.

Now I introduce the emergent algebra sector (over the set A).

Definition 4.  EMER(A)  is the subset of GRAPH (over the group \bar{\Gamma})  which is generated by the following list of gates:

  • arrows and loops,
  • \Upsilon gate and the gates \bar{\varepsilon} for any \varepsilon \in \Gamma,
  • the approximate sum gate \Sigma_{a} and the approximate difference gate \Delta_{a}, for any a \in A,

with the operations of linking output to input arrows  and with the following list of moves:

The set EMER(A) with the given list of moves is called the emergent algebra sector over the set A.

As you notice, I have not included the approximate inverse into the list of generating gates.  That is because we can prove easily that for any a \in A we have inv_{a} \in EMER(A).  (If \varepsilon \in \Gamma then  we trivially have inv_{\varepsilon} \in EMER(A) because it is constructed from emergent algebra gates decorated by elements in \Gamma, which are on the list of generating gates.) Here is the proof: we start with the approximate difference \Delta_{a} and with an \Upsilon gate and we arrive to the approximate inverse inv_{a} by a sequence of moves, as follows:

approx_op_5

We proved the following relation for emergent algebras: \Delta^{x}_{a} (u,x) = \, inv^{x}_{a} u.  This relation appears as a computation in graphic lambda calculus.

For another example of a computation in the emergent algebra sector see the post Emergent sums and differences in graphic lambda calculus.

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