I love your diagrams, Michael -- I think they make it much more clear how exactly wiring diagrams capture the kinds of things we can do with monoidal preorders!

> It might be because of poetic justice, but I think this may not be true? When you draw, there is an implicit order to the placement of arrows, circle or whatever it is you are drawing. So the following two pictures are actually not equal until you give it that rule!

I think that, since associativity does not change the order of wires at all, this does not matter. Your example shows reordering the red arrow with respect to the other arrows, but this is something commutativity does, not associativity. As John notes, the very idea of a wiring diagram embeds associativity into its core -- so much so that although you might expect there to be something explicit to say about associativty, there just isn't anything much at all!

For instance, consider your new associativity diagram:

> ![](http://aether.co.kr/images/monoidal_preorder_associativity.svg)

This doesn't actually capture associativity either, because associativity doesn't say anything about the kinds of relationships in the preorder. In other words, you're involving at least three, probably four(!) "reactions" in this diagram, when the associativity rule \$$(x \otimes y) \otimes z = x \otimes (y \otimes z)\$$ involves no reactions at all. Truly, associativity just says there is no distinction to be made by how you cluster your wires pictorially.