# Fan-out moves: CO-COMM, CO-ASSOC, GLOBAL FAN-OUT, LOCAL FAN-OUT

This is part of the Tutorial: Graphic lambda calculus.

Here are described the moves directly related to the $\Upsilon$ gate.

• CO-COMM move.  This is the local move depicted in the following figure

It means we may permute the outputs of a $\Upsilon$ gate. The name means “co-commutativity”, because the diagram resembles to the one of the commutativity property, with the exception of the arrows orientations, which are in opposite directions (hence “co-“).

• CO-ASSOC move. This is a local move, described by the next figure

It has the following effect: by using CO-ASSOC moves, we may move between any two binary trees formed only with $\Upsilon$ gates, with the same number of output leaves. The name means “co-associativity” and the explanation is similar to the previous one, with “commutativity” replaced by “associativity”.

•   GLOBAL FAN-OUT. This is a global move, because it involves a modification of an arbitrary number of nodes (gates) and arrows.

Precisely, the move acts like in the following picture: if $A$ is a graph in $GRAPH$  then we may replace the graph $A$ connected to an $\Upsilon$  gate by two copies of $A$.

GLOBAL FAN-OUT implies CO-COMM  (namely two GLOBAL FAN-OUT moves have the effect of one CO-COMM move).  The move CO-COMM is not useless though: we may choose to work in a sector of the graphic lambda calculus which uses CO-COMM but not GLOBAL FAN-OUT.

There is a variant of this move which is local.

• LOCAL FAN-OUT. Fix a number $N$ and consider only graphs $A$ which have at most $N$ (nodes + arrows). The $N$ LOCAL FAN-OUT move is the same as the GLOBAL FAN-OUT move, only it applies only to such graphs $A$.

LOCAL FAN-OUT does not imply CO-COMM.

Important remark: The gate $\Upsilon$ is really a fan-out gate only in the sense described by the FAN-OUT moves. In the absence of one of this moves, the gate cannot  be described as a “fan-out”. The “fan-out” is in the moves, not in the gate.

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