I've thrown together a (rather poor, compared to yours) diagram illustrating the distinction I'm trying to make. On the right side is the situation as we have it now, with a reaction \\(w \le x \otimes y \otimes z\\) assuming associativity.

![](https://i.imgur.com/W7fTaPH.png)

If we don't have associativity, then there are two possible reactions we might have meant. Let's suppose we want \\(w \le (x \otimes y) \otimes z\\) to be our reaction. We need to be able to distinguish between \\((x \otimes y) \otimes z\\) and \\(x \otimes (y \otimes z)\\), since the latter isn't a valid input to our reaction. So we have to represent the input differently; in this case, with explicit \\(\otimes\\) combiner nodes and reducing all reactions to single-input single-output representations.

In other words, if you replaced \\(\le\\) with \\(\otimes\\) in this diagram (and added the missing output edges), it would be completely accurate:

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

But since we bake associativity into our rules for working with string diagrams (namely, allowing multi-input multi-output relations), we never actually need the \\(\otimes\\) node. It's an entirely redundant symbol, since it's been incorporated into the representation of our reactions.

![](https://i.imgur.com/W7fTaPH.png)

If we don't have associativity, then there are two possible reactions we might have meant. Let's suppose we want \\(w \le (x \otimes y) \otimes z\\) to be our reaction. We need to be able to distinguish between \\((x \otimes y) \otimes z\\) and \\(x \otimes (y \otimes z)\\), since the latter isn't a valid input to our reaction. So we have to represent the input differently; in this case, with explicit \\(\otimes\\) combiner nodes and reducing all reactions to single-input single-output representations.

In other words, if you replaced \\(\le\\) with \\(\otimes\\) in this diagram (and added the missing output edges), it would be completely accurate:

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

But since we bake associativity into our rules for working with string diagrams (namely, allowing multi-input multi-output relations), we never actually need the \\(\otimes\\) node. It's an entirely redundant symbol, since it's been incorporated into the representation of our reactions.