# Dictionary from emergent algebra to graphic lambda calculus (III)

Continuing from   Dictionary from emergent algebra to graphic lambda calculus (II) , let’s introduce the following macro, which is called “relative $\Upsilon$ gate”:

If this macro looks involved, then we might express it with the help of emergent algebra crossing macros, like this:

Notice that none of these graphs are in the emergent algebra sector, a priori! However, by looking at the graph from the  left, we recognize an old friend: a chora.  See arXiv:1103.6007 section 5, for a definition  of the chora construction, but only in relation with emergent algebra gates. Previously the chora diagram appeared as a convenient notation for the relative dilation gate, but here we see it appearing in a different context. It is already present in relation with lambda calculus in the lambda-scale article arXiv:1205.0139  , section 4.  It will be important later, when I shall give an exposition of the second paragraph after Question 1 from the post Graphic lambda calculus used for quantum programming (Towards qubits III) .

With the help of the relative $\Upsilon$ macro, we can give a new look at the mystery move from the end of dictionary II post. Indeed, look at the following succession of moves:

The use of the mystery move is to transform the relative $\Upsilon$ gate into a usual $\Upsilon$ gate! So, if we accept the mystery move for the emergent algebra sector, then the effect is that the relative $\Upsilon$ gate is in the emergent algebra sector of the graphic lambda calculus and, moreover, it can be transformed into a $\Upsilon$ gate.

With this information let’s go back to the graphs $A$ and $C$ from the dictionary II post.

We know that $A \leftrightarrow C$ in the emergent algebra sector is the right translation of the approximate associativity from emergent algebras (formalism using binary trees, for example). In the last post we arrived at the conclusion that we can prove $A \leftrightarrow C$ by using the mystery move as a legal move in the emergent algebra sector of the graphic lambda calculus (in combination with the other valid moves for this sector, but not using GOBAL FAN-OUT).

Instead of the graph $C$, we may  introduce the graph $C'$ and compare it with the graph $C$:

The only difference between $C$ and $C'$ is  given by the appearance of the relative $\Upsilon$ gate, in $C'$, instead of the regular $\Upsilon$ gate in $C$.

We can prove  that  $C' \leftrightarrow A$. In particular this proves that $C'$ is in the emergent algebra sector, even if it contains a relative $\Upsilon$ gate. Recall that  if we don’t accept the mystery move in the emergent algebra sector, then it’s not clear if it belongs to that sector.  The proof is not given, but it’s straightforward, using the moves CO-ASSOC, CO-COMM, LOC PRUNING, R2 and ext 2 (therefore without the mystery move).

We can pass from $C'$ to $C$ by using the fact that the mystery move allows to transform a relative $\Upsilon$ gate into a regular one. So this is the way in which enters the mystery move in the equivalence $A \leftrightarrow C$: through $A \leftrightarrow C'$ , which can be done without the mystery move, and $C' \leftrightarrow C$ by the mystery move, under the form of transforming a relative gate into a regular one.